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POLICY RESEARCH WORKING PAPER 1523

Does More for the Poor Attempts to achieve 'more forthe poor" through the use of

Mean Less for the Poor? indicator targeting may in fact

mean less for the poor. The

The Politics of Tagging efficient use of a fixed budget

for poverty reduction may

require targeting. However,

Jonah B. Gelbach the use of indicator targeting,

Lant H. Pritchett using fixed characteristics that

are correlated with poverty to

determine the distribution of

expenditures, will tend to

reduce the budget. Ignoring

the budget reducing effects

can reduce the welfare of the

poor as they receive a greater

share of a shrinking budget.

There are political economy

limits to not only the scope

but the form of redistribution.

The World Bank

Policy Research Department

Poverty and Human Resources Division

October 1995

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POLIcl(:y RFSEARCH WORKINCG PAPFR 1 52)

Summary findings

Proposals aimed at improving che welfare of the poor * A classical social-choice analysis - in which agents

often include indicator targeting, in whichl non-incomne vote simultaneously about the level of taxation and the

characteristics (such as race, gender, or land ownership) degree of targeting - shows that positive levels of

that are correlated with income aret used to target limited targeted transfers will nor exist in equilibrium (an

funds to groups likely to include a concentration of the unsurprising finding, given Plott's 1968 thieorem). It also

poor. shows that a voting equilibrium often will exist with no

Previous work shows that efficient use of a fixed targeting but with non-zero taxation and redistribution.

budget for poverty reduction requires such targeting, * In a game in which the policymaker chooses the

either because agents' income cannot be observed or to degree of targeting while voters choose the level of

reduce distortionary incenitiv's arising from redistributive taxation, the redistributive efficiency gains from tagging

interventions. may well fail to outvweigh the resulting reduction in funds

Inspite of this, (ielbach and Pritchett question the available for redistribution.

political viability of targeting. After constructing a mnodel These results may be extended readily to account for

that is basically an extenlsion of Akerlof's 1978 model of altruistic agents.

"tagging," they derive three main results: Gelbach and Pritchett stress that even when these

Akerlof's result continues to hold: that, ignoring results hold, the alternative to targeted transfers - a

political considerations, not only will targeting be universally received lump-sum grant financed through a

desirable but recipients of the targeted transfer will proportional tax - will nonetheless be supported

receive a greater total transfer than they would if politicallv and will be quite progressive relative to the

targeting were not possible. pretransfer income distribution.

This paper - a product of the Poverty and Human Resources Division, Policy Research Department - is part of a larger

effort in the department to understanid the role of targeting in poverty alleviation efforts. Copies of the paper are available

free from the World Bank, ISIS H Street NW, Washington, DC 2043.3. Please contact Sheila Fallon, room N8-031,

telephone 202-473-8009, fax 202-522-1 1153, Internet address sfallon(ajworldbank.org. October 1995. (52 pages)

The Policv Research Working Paper Series disserinates the findings of work in progress to encouraige the exchange of ideas abott

development issues. An obhective o/ the series is to get the findings out quickly, even if the presentations are less thant fully polished. Ihe

papers carry the names of the authors and should be used and cited accordingly. The findinigs, interpretations, and conclusions are the

authors' ounp7 and should not he attributed to tbh World Bank, its Executive Board of Directors, or any of its memiber countries.

Produced by the Policy Rescarch Disseminiiationl Center

Does More for the Poor Mean Less for the Poor?The Politics of Tagging'

Jonah B. Gelbach and Lant H. Pritchett

1Gelbach is on leave from M.I.T. and acknowledges research support from the National Science Foun-dation via the Graduate Research Fellowship program. Pritchett is at The World Bank. We would like tothank Michael Kremer, Martin Ravallion, Paul Glewwe, Emannuel Jimenez, Estelle James, Dominique vande Walle and participants at seminars at the World Bank and M.I.T. for useful comments. Deon Filmeralso gave us indispensable help. The opinions expressed in this paper are solely those of the authors and donot necessarily represent those of The World Bank or the N.S.F. Correspondence to Pritchett by hardcopyat The World Bank, PRDPH, 1818 H St., Wash, DC, or electronically via lpritchett6worldbank.org.

Does More for the Poor Mean Less for the Poor? The Politics of Tagging

1 Introduction

Fiscal conditions requiring reductions in government expenditures often produce a dialoguelike the following between a policymaker (PM) and a technocrat economist (TE):

TE: We must reduce the deficit. Therefore, spending on category X (e.g.social security,telephone subsidies, education, health, irrigation water) must be cut.

PM: But if we cut the budget for X, the poor will be hurt, since they benefit fromgovernment expenditures on X programs.

TE: Don't worry. The current incidence of benefits from X expenditures is uniform,regressive, or at best only modestly progressive. Therefore, if we restrict X spending to thepoor, we can both cut the X budget and reduce poverty.

PM: But the administrative costs of identifying the poor correctly would be prohibitive.TE: Don't worry. Even if we can't observe actual income, we can find easily observable

indicators-like region of residence economic sector, occupation, gender, ethnicity, or land-owning status-that are correlated with household income. So long as these indicators arehighly enough correlated with income, we can target our X-expenditures to people basedon these indicators and still do more with less.

PM: But you assume that the budget for X will remain fixed. If I narrow the incidenceof X-related benefits by concentrating spending on targeted households, I also narrow theconstituency for spending on X. In fact, the ultimate outcome will be further cuts in thebudget X sufficient to wipe out any efficiency-related poverty reductions resulting fromtargeting.

Who is right? The theoretical foundation for the technocrat economist's argument wasprovided by George Akerlof in his 1978 paper, "The Economics of Tagging As Appliedto the Optimal Income Tax". Akerlof develops a strong and coherent argument for thepossibility that conditioning income transfers on immutable characteristics of potentialrecipients allows a more efficient social welfare system. The essential problem solved-or atleast mitigated-by the use of tagging (Akerlof's term for indicator targeting) is the tradeoffbetween the incentive costs of distortionary marginal income taxes and the social welfaregains of income or consumption redistribution. As the technocrat economist notes in thedialogue above, targeting is possible when some indicator set I can be used to conditiontransfers to recipients, so that when a person scores highly on some criterion C(I), sheis 'targeted" to receive a transfer (in addition to any other untargeted transfers she mayreceive). When these indicators are not easily manipulable, a targeted transfer will beessentially a lump-sum transfer for those who receive it, and the advantages of such transfers(as opposed to those conditioned on manipulable characteristics) are well known.2 AsAkerlof shows in what he calls the "rudimentary Mirrlees-Fair" setting (discussed formallybelow), when lump-sum targeting is possible, transfers to targeted individuals exceed both

2 1n industrialized nations, where income is observable (for the most part), these advantages largely derivefrom the fact that lump-sum transfers reduce or eliminate incentive distortions. As the policymaker in ourdialogue points out, in LDCs the issue is just as often the unobservability of income.

1

transfers to the untargeted and transfers in a world in which no targeting is possible. Akerlofpoints out that general theorems concerning the relative merits of targeting are not possiblein a world in which indicators may be affected by agents' behavior. We follow Akerlof inignoring this problem: as he notes, its treatment simply makes the issue an empirical, ratherthan theoretical, in character.

We laud the motives underpinning proposals to use indicator targeting. 3 However, thepolicymaker's objections may have merit: we believe that serious questions exist concerningthe political viability of targeting. In the rest of this paper, we develop a model thatessentially generalizes Akerlof's. The two primary differences are that we introduce athird, middle-income, class of agents (Akerlof has only high- and low-income people) andallow for "unemployment' of the poor and middle class. Adding a third income class ofcourse is necessary for any consideration of political issues. The introduction of individualincome uncertainty via unemployment reflects our belief that insurance motives play asignificant role in the determination of the degree of income redistribution in a society. Theother possible determinant is altruism. While the model can get along perfectly well whenaltruism is accounted for, we believe that altruism in fact is not a primary determinant ofredistributive politics. We offer a full section below treating both theoretical and empiricalconsiderations related to this issue.

We make one critical methodological assumption. While our use of a continuum ofagents rules out aggregate uncertainty over the number of people in a given income classreceiving targeted transfers, we assume that from the point of view of any given person,the event of receiving a targeted transfer is random. Without this assumption (and inthe absence of altruistic motives for redistribution) the politics of targeting is trivial: ifthe set of agents receiving targeted transfers has more than half the political power in thesociety, maximal targeting will be observed, while no targeting will be observed otherwise.Of course, this situation is uninteresting from a modeling point of view, but we also believethat it fails to reflect a sizable number of actually proposed targeting systems, particularlyin developing countries.4

3Hereafter all references to 'targeting' refer only to the type of indicator targeting we define formallybelow.

4As for developed countries, e.g. the U.S., our formulation may be les realistic, primarily because weasume independence of the processes generating bad income shocks and receipt of the targeted transer. Infact, one is eligible for food stamps only if one passes income and asset tests, which quite certainly will becorrelated with shocks to current-period income, like unemployment. The important fact is that it is notcorrect to sy that all targeting is deterministic, i.e. that people always know whether or not they will receivetargeted transfers. Some people of course will know that they never will receive such transfers: men in theU.S. do not receive AFDC. But as Congres moves toward ending the entitlement status of myriad targetedtransfers, it will no longer be true that targeted people know that they will receive targeted transfers. Inany case, we do not believe that taking these considerations into account is worth doing here, both becausewe focus in this paper on LDCs and because such a modification actually would make our analytical casestronger, since it would reduce the number of states of nature under which targeting benefits the middleclass.

2

1.1 Politics and Indicator Targeting

We take two distinct approaches in modelling political equilibrium. The first uses a classicalsocial choice model, in which binary elections are held between pairs of alternatives, whichare ordered pairs specifying a tax rate and the percent-ge of the budget allocated to targetedtransfers. In this model, an equilibrium is defined in the traditional way: a point z is anequilibrium if and only if no other point y is strictly preferred to x by a majority of voters5 .In section 4 we derive two basic results using this equilibrium concept. First, we show thatno equilbrium can exist with positive levels of the targeted transfer.6 This result is entirelyunsurprising in light of Plott's (1968) well-known and restrictive conditions for majorityvoting equilibria. However, our second social choice result is quite a bit stronger: underplausible conditions, no targeting is an equilbrium of the voting model. We stress that inthis equilibrium, a (possibly sizable) universal transfer will exist, enabling redistribution.

The force of this result depends on the subtle but critical distinction between the fol-lowing two statements: 1) 'No point y x z is an equilibrium"; 2) "The point z is unbeatenin majority voting by all points y # z (i.e. x is an equilibrium)". This distinction is criticalbecause it may well be (indeed it is often) the case that no voting equilibria exist. In suchcases, non-equilibrium status hardly is a reasonable criticism of any point, since all otherpoints have the same flaw!' However, when an equilibrium does exist, political actors floutit at their peril, so that economic advisers (like TE in the dialogue above) who suggest thatpolicymakers abandon an equilibrium position risk being ignored altogether.8

Our second approach is to imagine a game played between the policymaker and theelectorate, with simple Nash equilibrium being our solution concept. The policymaker isallowed to choose the proportion of the budget spent on targeted transfers, and then anelection is held to determine the level of taxation conditional on the policymaker's choice.We make use of the well-known median voter result, which is appropriate in this contextsince once the degree of targeting is fixed, only one dimension of policy (the level of thebudget) remains to be sorted out through the political process9. Within this median voterapproach, we distinguish two possible versions of our game. In the first, the policymakertakes the evel of taxation as given. ln the second, she takes account of political constraintsby taking afunction r-(k) as given, where k is the proportion of the budget spent on targetedtransfers and T is the function describing the median voter's optimal tax conditional on

"As discussed in Section 4, we weaken the defitition of equilibrium to requie a majority of voting power,rather than of voters, to allow ior the likelihood that political power is distributed non-uniformly throughoutthe population

'As stated in the text, this result actually is a special case; the general result, discussed in Appendix 2,is that there is sero probability that a majority voting equilibrium exists with positive levels of the targetedtranfer.

"This remark is even more appropriate in light of McKelvey's (1978) remarkable demonstration thatwhen the equilibrium set is empty, all points belong to the same cycle met, so that there is likely to be nosocial choice-theoretic basis for eliminating any points from consideration.

'Provided, of course that other political agents may propose the equilibrium point. We should note,moreover, that when no targeting is an equilibrium, smal perturbations of preferences do not in generalruin its equilibrium status, despite Plott's result that when interior equilibria exist (in our context, this meansequilibrium at positive levels of targeted transfers) they are not robust to smali changes in preferences.

'The other principal requirement for use of the median voter theorem, single-peaked preferences, willfollow trivially from our mumption that agents are risk averse.

3

k (we show below that this approach is mathematically proper). It is trivial to show thatwhen the policymaker takes account of political constraints, i.e. in the second version, shenever does worse-and almost always does better-than when she does not.

Moreover, we show that it is quite possible for the optimal politically-constrained level oftargeting to be zero-even though casual observation (e.g. large differences in income acrossregions, ethnic groups, occupations) may point to significant social welfare improvementsfrom positive levels of targeting using available indicators. This result has perhaps the most"economic" intuition of our findings. When people are unlikely enough to receive targetedtransfers, positive levels of targeting may be thought of as a tax on the activity of publiclyfinanced consumption insurance. Since in our model the amount of consumption shiftingdone is a monotonic transformation of the level of taxation, we can think of taxation as the"good" being "purchased". Hence for people unlikely enough to receive targeted transfers,targeting may be thought of as a tax on-or price increase of-the "good" of taxation. Thequestion of how the level of taxation changes when the degree of targeting changes thusboils down to the standard tug of war between substitution and income effects: whilethe higher price of unemployed- relative to employed-state consumption makes employed-state consumption relatively more attractive at the margin, it also makes a given amount ofunemployed-state consumption more expensive, requiring higher expenditures on the latter.Since we are operating in the expected utility framework, the crucial "parameter" of interestin resolving the substitution-income effect question will be the elasticity of substitution ofma.rginal utility, which is also the coefficient of relative risk aversion, across states of theworld. We often will use CRRA preferences, though this assumption is not critical (bychanging the word 'parameter" to "function", we can treat general preferences). When thedegree of relative risk aversion is (relatively) small, the substitution effect prevails, withthe result that the politico-equilibrium tax rate (and hence the overall budget for transfers)falls as the level of targeting rises. In this situation, it is possible for the optimal policy tobe no targeting of transfers: the efficiency gains from targeting may be outweighed by anasociated loss in the budget for redistribution.

Both of these results regarding the political viability of targeting stand in stark contrastto Akerlof's proposition. As he notes, his model is highly stylized, so that one might betempted to argue that the world described in our model-even in the absence of politicalconstraints-simply does not characterize one in which targeting would be useful anyway. Infact, as we show in Section 3, with only one very weak and natural assumption, Akerlof'sfirst result-that some targeting always is optimal-generalizes directly to our model. Thisfact is intuitive, for the basic reason that more instruments gives the policymaker morefreedom in designing policy; a small amount of targeting thus almost always brings anefficiency improvement, verifying the generality of Akerlof's result (in a non-political world).

1.2 LDCs and Indicator Targeting

In LDCs, indicator targeting is proposed not only as a means to reduce incentive problems(the focus of Akerlof's model), but also because conditioning transfers income simply isinfeasible. In one common type of indicator targeting, a limited data set (say, a house-hold survey) is used to establish a relationship between some readily observable household

4

characteristics (e.g. region of residence) and poverty (or income or consumption). Thisrelationship then can be used to condition transfers to the enitire population based on theidentified characteristics rather than on actual income.

For instance, in a particularly interesting empirical application of indicator targeting,Glewwe (1990) finds that the poverty minimizing policy in urban Cote d'Ivoire is to giveall transfers to households in the East Forest region. Of course, seen through an ez postlens, this proposal implies that no household not located in the East Forest-no matter whatits actual income-would have any chance to receive a transfer, and hence would have noself-interested incentive to support the proposal. As Glewwe stresses, this example paintsa particularly stark picture of the obvious and fundamental political economy probleminherent in deterministic indicator targeting ex post of the decision on the indicators andthe criterion function

Our argument is deeper: we show that even ex ante, before the criterion function C(I)is known, targeting may have adverse political economy consequences. For example, in anelection between two parties, one campaigning on a platform involving more focused (i.e.targeted) delivery of benefits and the other proposing the opposite, C(I) need not be knownfor the targeting party to be defeated. Alternatively, a party in power for the forseeablefuture might be forced to reduce overall redistributive expenditures in order to increasetargeting." 0 In either case, political opposition is to the possible outcomes of targeting.But if the "wrong" party is elected, or if budget cuts are large enough, targeting proposalsmay result in lower realized welfare for the poor. Moreover, insofar as parties that generallysupport targeting also support other economic reforms thought to be beneficial, proponentsof these other reforms should be sensitive to the practical possibility that political lossesresulting from targeting proposals may not be separable from these other reforms. Thepossibility that proposing targeting will have serious political consequences is no less realthan are these consequences themselves.

The rest of this paper procedes as follows. In Section 2, we present a brief summaryof Akerlof's model and his result. We then describe the components of our model. InSection 3 we discuss the optimal transfer regime in a world where the policymaker need notworry about political support. Then in Section 4 we embed the model in a social choiceframework. In Section 5, we take a Nash equilibrium approach to the political economyof targeting, letting the policymaker choose the degree of targeting while the populationvotes over the tax rate conditional on the policymaker's choice. In Section 7, we discussboth theoretical and empirical issues related to altruism. In Section 6 we describe existingempirical literature (related to targeting in developing countries) and policy implications,and we argue that our model does a reasonable job of capturing the basic characteristics ofproposals made in this literature. In Section 8 we conclude. Three appendixes contain, inorder, proofs of several results of the basic model, a generalization of the results of Section 4,and some technical details regarding the introduction of altruism in section 7.

10We imagine, for example, a system of government in which an executive branch may determine thebasic form of transfer distribution, while 'the voters' or, more realistically, a legislative branch, determinesthe overall budget. This framework is developed in detail below.

5

2 The Model

We begin by summarizing Akerlof's model (using our notation), and then set up our ownmodel.

2.1 Akerlof's Model

There is a continuum of agents having measure one. Half the population is skilled, while theother half is unskilled. There are two kinds of jobs, easy and hard, and they pay it and r,respectively; unskilled workers can do easy jobs only, while skilled workers can do either job.Utility of income is given by a twice differentiable, increasing and strictly concave functionu. Easy jobs carry no disutility of labor, but associated with hard jobs is an effort cost c,which enters the utility of hard-job workers additively. Hence in the absence of governmentintervention, unskilled workers have utility u(it), while skiUed workers have utility

u(r) - c, if the hard job is chosen

u(p), if the easy job is chosen

So long as u(r) - u(p) > c, skilled workers take hard jobs. We now introduce lump-sumtaxation and transfers. Letting T, and Tu be the tax (transfer) for skilled and unskilledworkers respectively, in any equilibrium with positive levels of T and Tu, we have the incen-tive compatibility constraint that u(r - T.) - u(i + Tu) > c and the budget constraint thatT = Tu (since the two types of workers have equal population sizes). It is straightforwardto show that when the policymaker's objective is to maximize the sum of utilities, bothconstraints bind in equilibrium, so that everyone has equal utility at the optimum.

Akerlof then introduces targeting in the following way. Some proportion a of the pop-ulation of unskilled workers is assumed to be targeted, so that the government may makeseparate lump-sum transfers to these people. The policymaker now has the instruments t8,tu,n and tut, which are (respectively) the tax on skilled workers, the tax on (if negative)or transfer to (if positive) unskilled, non-targeted workers, and the transfer to unskilled,targeted workers. The incentive compatibility constraint is now u(r - t,) - u(y + tun) > C;

we use tun rather than tug because skilled workers are untargeted. It is in this sense thatthis form of targeting is lump-sum in nature: no matter what they do, there is no way foruntargeted people to obtain the targeted transfer. The budget constraint (when incentivecompatibility is satisfied) is now t8 = 6tut + (1 - 6)tun. Letting asterisks denote optimalvalues, Akerlof's achievement is his demonstration that if u(r) - c > u(A),

1) tut > tun and2) tut > Tu.In words, l) says that the optimal transfer to targeted unskilled workers is larger than

the transfer to untargeted, unskilled workers, while 2) says that the optimal transfer totargeted workers exceeds the transfer they receive when targeting is not possible. This isprecisely the sense in which targeting means 'more for the poor".

However, both results suggest the presence of a significant political economy problem:the use of targeting drives a wedge between the two groups of poor agents, both of which

6

otherwise would support some form of redistribution rather than none. This point is mademost stark by Akerlof's observation that it is possible for t,u,, to be negative , so thatuntargeted, unskilled agents pay a tax; why would people expecting to be taxed under atargeted regime support that regime over an untargeted one in which they would receivebenefits? We now describe the model we use to consider this question.

2.2 Our Model: Description

The population in our economy has mass one, and there are three classes of people: rich,middle income, and poor; we use the subscripts r, m, and 1, respectively, to refer to them.Each group has a probability distribution over employment status and a correspondingmaximum marginal product when employed. For simplicity, we assume that the probabilitydistribution for the poor and middle income groups is the samell: with probability p > 0,these agents wiil receive zero pre-transfer income, and with probability q = 1 - p, they wiUreceive income according to the jobs they choose. We assume that the respective maximummarginal products of members of the rich, middle income, and poor groups are r-, 1, andp, with r' > 1 > p. An agent first finds out her employment status and then, if employed,chooses a job to work in. We assume that for each maximum marginal product level, thereis a corresponding job paying that marginal product as a wage, and there are no other jobs.An agent can choose to work in any job whose wage does not exceed her maximum marginalproduct. Thus rich people can work in any of the three jobs, middle income agents canwork in the two lower paying jobs, and the poor can work only in the lowest paying jobs.There is no disutility of labor, so that the only incentive compatibility issues we will haveto consider concern employed workers' choice of jobs.

We classify our three groups along two further dimensions: diversification of income andtax status. All agents working in jobs paying more than p must pay taxes, but those workingin p jobs need not."2 The rich (mass a,) have guaranteed income (i.e. can diversify), sothat if they choose to work in jobs that pay r, they will have income of r = qrr withprobability one. We can justify the guaranteed income of the rich either by assuming thatthey really are not subject to income uncertainty (q, = 1), or that they have access to anactuarially fair insurance market, in which case, so long as they exhibit some degree of riskaversion, rich agents will insure fully. For convenience, and without loss of generality, wewill put qr 1, so that r- = r. Middle class (mass a,m) and poor agents cannot diversify.

We define the tax base as y, so that y = o,r + qam, when everyone works in jobspaying their maximum marginal products. The assumption that y < q is needed below toensure that the tax chosen in a social choice setting is less than unity13 . If the rich work in

"The social choice results we will state are completely invariant to changes in this assumption, tlhoughour results on the optimality of positive levels of targeting are not. Nonetheless, the invalidation of theseresults would require pi < Pm, which seems counterintuitive.

12There is a potential consistency problem here: how is it that we assume people pay an income tax whenincome is unobservable? Any proportional tax, e.g. a VAT, would do; alternatively, we might think of thetax and transfer sides of government activity as unconnected, so that a revenue agency collects payroll taxeswhile a benefits agency pays them out, and the two agencies are unable (or unwilling) to match up records.In ny event, we believe that this issue is of minor importance.

13This assumption might seem objectionable, since it appears to say that pre-transfer mean income is

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r'-marginal product jobs but middle income agents work in s-marginal product jobs, thenwe will have y = a,r; if everyone works in yt-marginal product jobs, then y = 0, i.e. noone pays taxes. We will see below that the rich never will choose to work in middle classjobs, so we need not consider y for those cases.

Any transfers are financed by a linear tax r E [0, 1], and two kinds of transfers arepossible. The first is a universal, or nontargeted transfer N; everyone in the economyreceives this subsidy. The second is a targeted transfer, 6, which is received by only thepopulation proportion 6 = Eiaii,, i = I,m,r,, where 6i is the proportion of agents oftype i who receive 6. Given our assumptions of probabilistic targeting and a continuumof agents, bi also is the probability that any given person of type i receives the targetedtransfer. Throughout the paper, we will assume that 6, = 0, so that no rich agents receive6; thus 6 = a161 + cimm We stress that the policymaker is permitted to condition targetedtransfers neither on realized income level nor on employment. This assumption capturesthe policymaker's first objection (in our opening dialogue) to targeting, namely that it isnot feasible for administrative reasons. However, since 6 >6 m >0 -6,, using targetedtransfers allows the policymaker to avoid giving benefits to the rich, who are known not to"need" them, while directing them to the poor and middle class, at least some of whom doneed them. This fact makes targeting more efficient in a non-political world."4

We do not endogenize within the model the particular way in which the the b's might begenerated under a particular proposal for indicator targeting; whether or not regressions (orsimple cross-category income data) based on surveys or other techniques are used is not acentral issue. In practice, what is central is that the "indicator" used be (relatively) perma-nent and not depend on observing current income status. Were temporary characteristics-inour model, employment status-observable and hence "targetable", we would have a target-ing technology with four 6s, two for each of the poor and the middle class. In a loose sense,the setup we use captures the permanent-income characteristics aspect of most (developingcountry) targeting proposals, e.g. using gender, ethnicity, disability, or other "long-term"characteristics to target recipients.

We summarize the model's structure in Table 1. For each of the three groups of agents,the table shows their maximal marginal products, their tax status, their probablity of a"bad shock," and their targeting probabilities.

less than pre-transfer median (expected) income. However, when everyone works at her maximum marginalproduct, pre-transfer mean income actually is greater, since it is defined as y + ugq. Moreover, pre-transfermedian income will depend on the population proportions (the e,) and the probability of unemployment;while it is possible for this median to be less than p, unless p is very large it will be any number m E (,. 1).

' 4 0f course, in this context "efficiency" means that the benefits to the winners outweigh the costs to thelosers, of whom some will always exist: with 6a or 6

m, less than unity, targeted transfers are not provided tosome poor and middle class agents who need them.

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Table 1: Structure of the modelType of agent

Poor Middle RichPopulation share al am a,Income ifemployed L I rProbability of"unemployment" p p 0Can Diversify No No YesTax Rate 0 T rUniversal transfer N N NProbability of 6E l m 0receiving targetedtransfer = 6Utility function Von nNeumann Morgenstern utility function u:

u' > 0, u" < 0.

2.3 Budget Balance and Incentive Constraints

Budget balance requires that N + 60 = yr. Defining k as the proportion of the budgetspent on targeted transfers, budget balance is expressed by the two identities

N (1 - k)yr(1)

The efficiency gain from targeting is directly attributable to the T in the denominatorof the first identity: a dollar spent on targeted transfers allows recipients to receive 1/6dollars, while a dollar spent on untargeted transfers gives only one doUar in transfers toeveryone in the economy.

It will be convenient to define y (1 - T)/6 and -yi similarly for each group i. We notethat N + = ( 1 + Tk)yr, and hence O(N + 0)/Ok = Tyyr and cl(N + 6)/Or = (I + Tk)y.

All agents have von Neummann-Morgenstern preferences with the twice continuouslydifferentiable state-contingent utility function u, u' > 0, u" < 0.15

Indexing employment status by h = 1 if an agent is employed and h = 0 if not, andindexing receipt of the targeted transfer by j = 1 if B is received and j = 0 if not, we candefine consumption for type i individuals in state hj as yi . Poor agents' possible incomestates appear in Table 2a.

For the middle class and the rich, we need to take account of job choice in determiningstate contingent post-transfer income. Hence we require that if, given the tax and transferregime, an employed agent could do better by earning a lower before-tax income, that agent

15Again, for our social choice results, we need not assume that the function u is the same for all threetypes of individuals; for our results on the (social) optimality of positive levels of targeting, we would notrequire identical u functions, but some restrictions would be necessary.

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will do so."6 Since everyone always receives N, its size is irrelevant for incentive compat-ibility concerns. Moreover, the fact that targeted transfers are conditioned on underlyingcharacteristics rather than on observed income means that even if, say, bm < 6i, the sizeof 0 is of no concern to middle class agents in their decision over which job to take whenemployed. It is for this reason, as Akerlof shows, that a targeted transfer may be desirable:unlike transfers conditioned on income level, fully lump-sum targeted transfers do not re-ward inefficient behavior."7 As such, for employed members of the middle class, the onlyfactor affecting which job to take is the relative size of their income when employed, 1- -r,and the income they would have if they were to masquerade as poor agents, which is A.

Hence we have our first incentive compatibility constraint:

(IC)m r < 1 - A (2)

Middle-income agents' state-contingent incomes are described in Table 2b.

Table 2a: Poor Agents' Income By Employment and Targeting StateUnemployed (h = 0) Employed (h = 1)

Do not receive 9 (j = 0) yOO = N YIO = N +AReceive 0 y,I = N+O Yll N + +

Table 2b: Middle Class Agents' Income by Employment and Targeting StateUnemployed (h = 0) Employed (h = I)

Takes Job Paying /i Takes Job Paying 1Donotreceive6(j=0) yo =N Ymo=N+ / ypm=N+I-rReceive 0 Yomo = N + 0 yo = N +6+it y N+ +1-r

The associated consumption state probabilities for type i individuals are irh*;, i.e. vr =p6 t, Troo = p(l - b6), etc. Ilence overall utility for an agent of type i, i = I,m, is U,(k,T) =

,h ,i 7rhju(yhj), h,j = 0, 1. To find the incentive compatibility constraint for the rich, wefirst note that since r > 1 and people working in middle class jobs pay taxes when employed,rich agents never will take middle income jobs. Hence the constraint that matters for therich is the one that keeps them from taking low income jobs. Again, neither N nor 6 canaffect this margin, so the constraint is simply

(IC)T r< 1 - (3)r

It is obvious that in any equilibrium with positive levels of transfers, (IC)r must besatisfied, since if (IC), is not satisfied, neither is ([C)m, and hence the tax base is zero. Wewill show below that in both thie policymaker's optimal solution with no political constraintsand the political equilibrium, positive levels of transfers will exist; as such, we require (IC),

"'t should be noted that this issue arises only because the poor pay no taxes, so that for the middle classand the rich, taking untaxed p-jobs is advantageous if the tax rate is too high.

"'Hence what we mean by targeting is precisely what Akerlof meant, and not what Nichols and Zeckhauser(1982) mean.

10

to hold throughout the rest of this paper. Hence, rich agents' post-transfer income is justy' = r(1- r) + N, and their utility in all states is u(yr).

Before concluding this section, we note that yhj > y 1-middle class people always doat least as well as poor people in the same employment-targeting state-and yho, yO • < yr

non-rich people who do not receive 0 never do better than rich people.

3 The Policymaker's Optimum With No Political Constraints

In this section, we ignore political constraints and explore the characteristics of the op-timal tax and transfer regime when the policymaker faces only the budget and incentiveconstraints. We will generalize Akerlof's first result directly: optimality requires the useof some targeting, i.e. k > 0. but also requires that targeted agents receive a greatertotal transfer than they would in a world where targeting was impossible. This finding isinteresting not only because our economy is more general than Akerlof's, but also becausewe consider a narrow set of feasible policies. Since Akerlof's economy has only two typesof agents and no income uncertainty, his tax and transfer system can be derived from ei-ther a lump-sum (once job choice is fixed) or linear tax system. In adding another typeof agent and unemployment, we force ourselves to choose between the two approaches; asthe enumeration in the previous section makes clear, we choose the latter approach. Henceour verification of Akerlof's results is not only an extension, but also a broadening, of thecircumstances known to foster the efficiency of targeting when political constraints do notoperate.

Since our economy has a continuum of agents at each consumption level, we are able todefine social welfare over realized consumption states (with each consumption state occur-ring in the same frequency as its population-proportion weighted probability). Assumingthat the policymaker's goal is to maximize some appropriately weighted, nondecreasing andconcave evaluation function G, the policymaker's objective function is

S(k,r) - a5lEr' G(u(yl )) + amZE1r'G(u(y'%)) + arG(u(yr)), (4)h j h j

and her problem when facing no political constraints is to

max{k,,)S(k,r) subject to budget balance(l)and incentive compatibility(3). (5)

We may now state our first result in Theorem 1: when enough poor people receive thetargeted transfer, the politically unconstrained optimal policy necessarily involves a positivelevel of targeted transfers.

Theorem 1 (Some targeting is optimal with no political constraints) Suppose that6g > 6 and G is differentiable. Then any (k*, r) solving A) will have k* > 0.

Proof: It is sufficient to show that for any tax rate, social welfare is increasing in k whenk = 0 (i.e. that 8(O,r) > OVr). We first show that this condition is satisfied when G is

11

the identity function, i.e. the policymaker maximizes a utilitarian social welfare function.Note that when k = 0 income is independent of targeting status for each income group,Y = Yhl for aUl i. Also note that when k = 0 the income of the poor and the middle classwhen unemployed is just the universal transfer, yI = = yT = N. Substituting in thoseequalities and differentiating, we have in the utilitarian case

as (O,r) = pyru'(N)[l - - (1- ar - a)]

+ qyrolu (lj)( -1) + amU,(Ylj)(6n-I)]-ar-yrul(yr)+ oy-r[alu,(YI) + 7t'Y')T- )3 7yU(T

Dividing through by yir, this expression can be rewritten as

pa,[u(N) - u'(y )] + q{ (61 - )[u-(yl U(YJ)] + Ur [u'(Y') - u'(Y3)]} (6)

Since N < y', yl < yl7, ym < yr, and u" < 0, all bracketed terms involving differencesin u'(.) quantities are positive; hence, 61 > i is sufficient for the entire term to be positive inthe utilitarian case. If G is not the identity function, then each u'(.) term is multiplied byan associated term G'(u(.)). Since G is concave, this operation only increases the relativedifferences in the u'(.) terms.Q.E.D.

The condition 6b 2 T is sufficient, but clearly stronger than necessary, since the otherterms in ( 6) are strictly positive. Moreover, this condition is very reasonable, since with

, -0, the condition may be rewritten as

i > n + v im, (7)am + a~r

from which it is clear that the condition simply requires that 61 not be too much less than6,m, whereas typically we would expect poverty-oriented proposals for targeting programsto have 6b > am: poor people should be more likely than middle class people to receivetargeted transfers.

One obvious result of Theorem 1 is that targeted people of a given income type receivegreater post-transfer income than do untargeted people of the same type. For our purposes,this finding is the important half of Akerlof's result.18 We now turn to the social choicecontext.

"'li addition, Akerlof demonstrated that so long as higher income agents' (IC) constraiat is satisfied whenthe tax level is zero, whea targeting is possible, recipients of the targeted transfer have greater post-transferincome than they would under the optimal policy when targeting is not possible (i.e. when k is constrainedto be zero). [i our terminology, Akerlof proved that I k>0 > yh;4A,,o, where asterisks denote optimality.While we have been able to derive restrictions on our model's non-preference paramneters under which thisresult holds here as well, they are not particularly intuitive. Moreover, without placing restrictions (beyondconcavity and monotonicity) on the form of the utility function, the conditions under which the result holdsare not very broad either. Since the result is not our main focus, we forgo a detailed statement of theseconditions.

12

4 Targeting and Social Choice

In this section we take a classical social choice approach and prove two basic results. Thefirst establishes conditions under which there can be no equilibrium with positive levelsof targeting, so that the only possible equilibrium entails no targeting. The second resultestablishes conditions under which an equilibrium with no targeting does exist. We stressagain that even with no targeting, there will be redistributive universal transfers. Afterintroducing some useful notation in the first subsection, we prove the theorems in thesecond.

4.1 Social Choice Framework

In an election between two proposed points, we will say that one beats another if it hasvoting support of more than half the population. Since the groups are continua, withinwhich all agents are identical (and hence will vote identically), we may think of the modelin terms of three representative agents, each having total voting strength vi, with E, vi = l.We do not impose any relationship between a group's population size and its voting strength.As such, we may consider societies where, for example, a large majority of people are poorbut form a relatively insignificant political force, a situation common to many developingnations. However, we do assume that no group is decisive, i.e. that vi < 1/2 V i; thisassumption is necessary for there to be a political economy issue at all. Once it is made,however, it is obvious that any combination of two groups forms a majority coalition, sothat our social choice analysis may be conducted in a traditional three-person majorityvoting framework.

For our purposes, the fundamental result in such a setting is Plott's (1968) theorem onthe existence of interior majority voting equilibrium. In the present context, Plott's twosubstantive conditions are that at any equilibrium point (k,r) > 0,

1) Members of one of the three groups must have maximal utility (for a local equilibrium,we require only that utility be maximal locally) and

2) The other two groups' marginal rates of substitution between k and T must be equal.We verify in the next subsection that these two conditions are mutually exclusive for

interior points. Ilowever, a fact that seems to have been overlooked in the social choiceliterature is that for non-interior points, e.g. points of the form (0, r), the second conditionis considerably weaker. So long as the two non-maximizing groups do not have their indif-ference curves intersect except at the interior point, existence of equilibrium is consistentwith unequal marginal rates of substitution, if the inequality is in the right direction. Insubsection 4.3 we offer intuitive conditions under which such an equilibrium will exist.

We now introduce some terminology that will be useful in our consideration of targetingin a social choice context. First, we denote the entire feasible set of tax rates and targetingproportions, [0, 1] x [0, 1], as F. At times it will be easier to carry out our analysis if we canconfine our attention to the set of values of r for which both (IC) constraints are satisfied.19

l9Since when r just exceeds I - i, employed mliddle income people switch into low-income jobs, sothat there is a discrete change in everyone's after-tax, after-transfer income; this complication means thatindifference maps, which we describe below, will not be continuous ay 1 - p.

13

Thus we define the set of points for which both (IC) constraints are satisfied as

(1) =_ {(k,r): k E [O,l|andr < I-j}

We then define

Bi(k,#) _{(k,r): U,(k,r) > Ui(k,?)}

Li(k,;) _ {(k,r): U,(k,r) = U,(&,r)

Respectively, these sets are group i agents' better-than and level sets.20

Now define the win set of any given point (k, ;) as the set of all points that is preferredto the point by at least two groups.

W(k,r) = {((k,r): (k,T) E (Bi(k, ,) n Bi(k,*)) for some i ¢ j}

We define the set of voting equilibria over some set X C_ as points that cannot bebeaten by any other points, that is, with empty winsets

E(X) {(k,r): W(k,r) n X = 0}

We will define a local equilibrium in X C Y as any point (k, r) E E(O n X) for someopen ball 0 around (k, r). When we want to consider the whole set of feasible (k,r), i.e.[0, 11 x [0, 1], we will simply write E for the associated equilibrium set. With this notation inhand, we begin our consideration of social choice issues by addressing the political viabilityof positive levels of targeted transfers when all of Y, i.e. the entire set (0, 1i x (0,1], isadmissible.

4.2 Targeting Cannot Exist in a Voting Equilibrium

2 1 Since the poor do not pay taxes and do receive transfers, it must always be the casethat DUIlOr > 0, which implies that poor agents always prefer a tax rate of unity to any

3OWe note that B. and L. define transitive partial orderiap for eachl group i, with their union defining a

complete transitive ordering. Using the standard terminology, we have

(k,r) E B,(k,t) * (k,r)P,(k,r)(k, r) E L(k ;) * (k, r)l,(lk ;)

(k,r) E (B,uL,)(k,t) * (k,r)R,(k,;),

where P., 1. and R. have their standard choice-theoretic meanings."In this subsection, we will assume that p = O, i.e. tiat the poor have zero marginal product. While obvi-

ously unrealistic, this assumption is made solely for ease of matlematical exposition. Given our assumptionthat poor agents' marginal product is zero, F(p) = r(O) _ . Quite obviously, under this assumption both(IC) constraints are satisfied trivially, so we have y o,r + ve,. AlU qualitative results iik this section areextended to tile general case p > 0 in Appendix 2, while non-intuitive or tedious proofs of results derivedin this section are relegated to Appendix 1.

14

other feasible value. Similarly, the rich always pay taxes, and since we have assumed thatr > I > y, the rich always pay more in taxes than they receive from N. Ilence it mustalways be the case that OU,/0r < 0, whiich implies that the rich always prefer a tax ofzero.

We now offer a fundaiiiental condition on 6m and T and use it below to prove Lemma 1,which establishes that when the condition is satisfied, middle class agents always prefer notargeting (and also always prefer any constant-tax reduction in targeting at any level oftargeting).

C 1

<m < 6

We note that this condition can be rewritten as 6,.. < a -61, so that it requires thata smaller proportion of middle class tlkan of poor people receive 0. We choose to writethe condition as in C I because in that form it emphasizes the comparison a given middleclass person, who has measure zero, makes between hier own chances of receiving B and theoverall number of people in the population receiving B. This approach is useful since thebasic logic of Lemma I below has to do with precisely this comparison, rather than onebetween the poor and middle class group probabilities of receiving B (it so happens that theindividual and group probabilities are identical, but that fact is irrelevant to the individualagent's assessment of various (k, r) points).

Lemma 1 (The middle class prefers less targeting when 6,, < 3) Suppose that C Iholds. Then OU,10/Ok < 0, with equality at k = 0 iff C I holds with equality.

Proof: The truth of the Lemma could be demonstrated algebraically by partial differenti-ation, but a more intuitive proof is possible. First, niote that for 6, < 1, using some of thebudget to fund a targeted transfer increases group-i agents' uncertainty about their income(above and beyond the exogenous uncertainity attributable to unemployment). Next, re-calling from equation ( I) that B _ kyr/b, the expected value (over targeting-receipt statesof the world) of income for an agent in group i (when employment status is fixed) is

i(y+ N +)+ (I- )(y'+N) = y'+N+69i

= y'+N+ +4kYr

6i= y' + yr(I - k(l-=i)) (8)

where y' is after-tax, pre-transfer income of agent i. Noting thiat I - 6jl6 > 0 by hypothesisof strict satisfaction of C 1, the expected value of incotmie over targeting states is decreasingin k. lhence when C I holds strictly, targetiing adds unicertainty to middle class incomiie anddoes so at a retura that is worse thlant actuarially fair. Replaccitiemet of > with = in the

15

appropriate places yields the result that targeting is actuarially fair for the middle classgroup when C 1 holds with equality. It is a well-known result that risk-averse agents willreject gambles that are not better than actuarially fair. Hence we must have OUi/ak < 0,with equality at k = 0 if C 1 holds with equality; but this is the statement of the Lemma.Q.E.D.

Thus satisfaction of C 1 implies that the unique local-and hence global-maximum of U,roccurs at (0, arg max, Um(0,r)). Moreover, the condition implies that (k,;) E Bm(k,f) X*

k < k. Similarly, since 6, = 0 trivially implies satisfaction of the analogue of C 1 for therich, we see that the unique local (and hence global) maximum of U,(k, r) occurs at (0, 0).Combining application of Lemma 1 and our result that OU,/19ir < 0, it must be true that(k, r) E B7,(i, r) if k < k, T < r, or both.

The level sets Li(k,r) will be of fundamental importance in our analysis. It will beconvenient to refer to them in terms of their associated indifference maps ri(k; Ui), whichmay be defined as that level (or set of levels) of the tax rate that keeps utility constant atUi when the degree of targeting is k.22

The only one of these three indifference maps that we may describe explicitly (withoutstrong assumptions on u, Em, or 61 ) is r,. Since rich voters face no uncertainty over income,their level sets are determined by those variations in k and r that leave them with constantconsumption, so that for any level of consumption c,, we have

7r,(k; u(cr)) -r (1- k)y (9)

Figure la depicts a typical map r, and the better-than set B, of any point lying on it.To continue, if we know that u(k,;) = u(c,), we can write

r,(k; u(c,)) - (r k)Y (10)r -k)y

Unless strong simplifying assumptions on the Es are made, it is not in general possibleto give closed-form definitions of rl and rm. IHowever, as we show below, the primary results

2 2Given values U° of the voters' utility functions, over some domain K,(Ui) of possible values of k, groupi's level set will have an associated function (for the poor and the rich) or correspondence (for the middleclas) mapping a valuc of k to values of r that yield the same utility to members of that group as does (k, r )The poor and the rich have functions because their utility functions are strictly increasing (the poor) orstrictly decreasing (the rich) in r, so that no value of k can be mapped to more than one value of r yieldingthe same level of utility. The middle class have an indifference correspondence because for any value of k,there may be zero, one, or two values of r yielding a given level of utility. (The possibility of zero is trivial,and by strict concavity of Urn in r, we know that there cannot be more than two values. There will be onevalue if either that value is a maximum (conditional on k) or a non-feasible tax (i.e. greater than I or lessthan zero) were necessary to achieve the given level of utility.) That said, we have r: K1(U) - [0,1],r, Kr(U,°) - [0, 1], and r.: Khm(Um,) => (0, 1] , where

K. (Uio) {k 3rs. t. U,(k,r) = U°O

16

we will need for Tl may be gotten with the aid of the implicit function theorem. Moreover,the only qualitative result we will need regarding middle class agents' preferences is theprevious lemma's, namely that when C 1 holds, middle class people will vote for constant-tax reductions in k at all points. All necessary intuition regarding rm may be gotten fromFigures lb, an example when C 1 is satisfied, and lc, in which it is not.

It will be useful to define a function r*(k) which satisfies

r'(k) = argmax Um(k,r)

T E [O,1]given I = 0 (11)

This function tells us the optimal tax rate, from the point of view of middle class agents,when the degree of targeting is k.23 The function is given implicitly by the equation

-m = 0 (12)Or

By the implicit function theorem, for r (k) E (0, 1) we have

dr i 2Um/..ksr (13)dk a2Um/0r 2

Similarly, we can define a function k*(r), which tells us middle class voters' optimallevel of k when the tax rate is T:

k'(r) = arg max Um(k,t) (14)ke(o,1]given p = 0

given by24

Um ,, (15)Ok

and whose first derivative, when the assumptions of the IFT are satisfied, is given by25

dk_ d2Um/Ok,9r (16)dTr 2Um/0k 2

We can now prove another lemma.

Lemma 2 (Conditions for the middle class to prefer r < 1) Suppose p = 0. Thenv < q, i.e. o,r < q(l - om), if and only if

(i) r(O) < I and(ii) At any point (k, 1), either OU,/Or < 0, OUm/0k < 0, or both.

23The opportunity set [0, 1] is compact and our assumptions on u imply that U, is continuous and concavein r over this set. Thus by the theorem of the maximum, r exists and is continuous.

24of course, when C I is satisfied, we have aUrn/ak < 0, so that k(r) = 0 and dk/dr = 0.25We note that by our assumption that u" < 0, the denominator terms in both first derivatives are

negative. Hence each first derivative has the sign of the cross partial derivative in its numerator.

17

t s~~~~~~~~~~~~~0

Co,oJ=rk KFigure la: IC Figue Ib:

Rich voters, in'ncc klidle cIAss vow!'

cwve c, ad ar tci i cure adbcttrhan set when

bawr4an set B,. 6, . Tg& opamtv

R;^ch vout have on mmbe s (O, V(O)) inopumaun ra(O,O)c

Fig=s I d:

Figu lc: mk lok ikM(dl cla vaters'

MmP mLcre an width bster-un svWbaurt lha set wrhena 8r 1 PoU== lies

6. Th&at e I * some/po /t

optunw mus lie a anIntc post lile m-.

Proof. See Appendix 1.

Conclusion (i) of Lemma 2 evidently implies that when j. = 0, the point (0, 1) is nevera voting equilibrium, since the optimal tax rate for both rich and middle class voters whenk = 0 is less than unity. For technical reasons, conclusion (ii) is useful below in ruling outthe possibility of positive levels of B in equilibrium (when j. = 0).

Next, we use Condition C 2 to establish in Lemma 3 an instrumental single-crossingproperty between ri and Tr. This property will rule points (k,r) E (0,1) x (0,1), i.e. pointsin the interior of X, out of the equilibrium set.

C 2

,>1- r

Lemma 3 (Single Crossing for rr & TI) If C 2 is satisfied, then(i) Any two indifference curves rT(k; U1) and Tr(k; c,) intersect at most once.(ii) If Tli(k) = rr(k), then rj(k) < (>)Tr(k)Vk < (>)k.

For k = 0, (i) and (ii) hold only if C 2 is satisfied.

Proof. See Appendix 1.

Figure Id depicts a typical example a poor agent's indifference map Ti. The singlecrossing result established in the previous lemma is depicted in Figure 2a. When the resultholds, any possible rich-poor coalitions to defeat a point are limited to points having lowervalues of k than the point in question. lIence b is beatable, but a is not, since it lies on thevertical axis, and negative values of k are not permitted. Figure 2b shows what happenswhen C 2 is violated: any point on the vertical axis is locally beatable by a rich-poorcoalition in favor of points like those in B1 n Br.

As for coalitions involving the middle class, Figure 2c shows an example in which themiddle class and the rich may form a successful coalition. In fact, Lemma 1 tells us thatso long as 6 m < 6, such coalitions always are possible at interior points (such as a). Figure2d depicts two possible situations regarding poor-middle class coalitions. The point a isbeatable by any point in the region I (which includes that part of the vertical axis adjacentto it), while the point b cannot be beaten by a poor-middle class coalition.

With these graphs and Plott's conditions in mind, any interior majority voting equilib-rium must resemble the situation shown in Figure 3. The point m' is optimal for middleclass voters (we include the locus of points rm only to remind readers what middle classvoters indifference curves must look like in this case; for any point on this locus, the ar-rows indicate the direction of improvement for middle class voters). At the same time,the better-than sets for rich and poor agents have to be disjoint, which implies that theirindifference curves are tangent, i.e. that rich and poor voters have equal marginal ratesof substitution at 7n'. Hence interior equilibrium requires the simultaneous violation of

18

41b~~~~~~~~~~~~~4

Kr s

%[W!til 9 sres31'''/

conditions C I and C 2. Lemma 4 shows that this situation cannot happen. In the theoremthat follows, we bring together this lemma and the ones preceding it to show that the setof interior equilibria is empty.

Lemma 4 At least one of C 1 or C 2 is satisfied.

Proof: Appendix 1.

Theorem 2 (Targeting cannot exist in equilibrium (Plott, 1968)) Suppose A = 0.For any set X C F satisfying int(X) $ 0 and any point (k, T) E int(X), W(k, T) is locallynonempty. (Equivalently, E(O n X) = 0 for all open balls 0 of (k, r) and (k, r) E int(X).)

Proof: We begin by demonstrating the result for F. By Lemma 4, no point (k, r), k > 0, cansatisfy both of Plott's conditions, since either the middle class and rich can form a coalitionin favor of a constant-tax local reduction of k or the rich and poor can form a coalition toreduce k locally while increasing r locally. This establishes the result for X F. But sincethe coalitions just described favor points that are arbitrarily close to (k,r), the result mustcarry over to any arbitrary X C F.Q.E.D.

This result is obviously negative from the point of view of supporting targeting, butby itself it is hardly sufficient to ruin the case for targeting. An enormous and well-knownliterature has established that social choice models do not in general have voting equilibria,so it might be the case that an equally negative result holds for any non-targeting (k = 0)regime. That is, it might be the case that while positive levels of targeting always arebeatable, there always exists a positive level of targeting that can beat any non-targetingregime. This possibility is exactly what we rule out in the next subsection.

4.3 Equilibrium Exists With No Targeting

We now prove Theorem 3, which gives necessary and sufficient conditions for the existenceof majority voting equilibrium with no targeting (we stress again that positive iransferswill occur in such an equilibrium).

Theorem 3 (Equilibria exist with no targeting.) Suppose bl > am. Then both C 1and C 2 are satisfied if and only if E(F) = {(O, r*(O))}

Proof See Appendix 1.

We point out that the results given so far imply that, while an equilbrium need notexist, if one does it can occur only in a non-targeting regime. HIence even if one wishes todispute the assumptions used in Theorem 3, one still must face the fact that no other pointin the space may be supported as an equilibrium.

Aside from noting that we have put j = 0 (an assumption we deal with in Appendix 2),two basic attacks on this theorem's assumptions are possible. First, one might argue that

19

t§}7w j t; 1tB,

4-,\

we should not assume 61 > 6m. We are perfectly willing to entertain the idea that when thisassumption is violated-which implies that 6m > a-targeting might be a good idea from apolitical point of view: our aim in this paper is to argue that targeting is politically viableonly when the benefits of targeting are dispersed over a large enough group of people. Buta targeting regime in which the middle class are more likely than the poor to receive atransfer is not what most people have in mind when they propose to target transfers.

Second, one might argue that the requirement in condition C 2 that 6, and 6m besufficiently close is overly restrictive. In response, we note first that while the condition isalways sufficient, it is necessary only at k = 0. Moreover, while we admit being less thansatisfied with the condition, it does have an intuitive explanation. When 61 is large bycomparison to 6 m, it also must be large relative to 3, so that the poor gain a large amountfrom targeting relative to taxation; that is, they are more willing to agree to tax reductionsin return for increased targeting. The rich, on the other hand, will be more willing to agreeto such deals when their taxable income is large relative to the transfers they receive, i.e.N. Recalling that the income of the rich always is r(l - r) + N, it makes sense that asr increases, so that the rich care relatively more about the taxes they pay than about thetransfers they receive, they will be more willing to accept more targeting in exchange forlower taxes. Ilence this attack on Theorem 3 relies on the claim that the rich and thepoor are likely to agree on proposals to cut taxes in return for more targeting. Whetheror not this phenomenon generally occurs in the real world is an empirical question worthanswering, but rarely (if ever) considered in existing empirical work. As a practical matter,however, the requirement that bi be sufficiently larger than 6 m requires relatively highperformance from the indicators used to develop the criterion function that is to identifythe poor, and most existing empirical work does not suggest such performance is possible.

5 A Nash Equilibrium Approach

5.1 Introduction

Our second approach to modelling the politics of tagging is to assume that elections are heldto determine the level of taxation, given the policymaker's choice of the level of targeting.This approach allows us to assess the applicability of the targeting asessment literature,where authors typically assume what we will call the "naive policymaker". That is, theseauthors ask the question, "For a fixed budget, what kind of targeting regime, based onobservable indicators, minimizes the level of a poverty index?"2 6

The Glewwe (1990) example cited in our introduction is an excellent example of thisapproach. Hle calculates that with a fixed budget of 10 million CFA francs, a uniform percapita transfer in Cote d'lvoire could reduce poverty by six percent. By contrast, usingestimates of the relationship between income and household characteristics to implementindicator targeting, that same 10 million francs could reduce poverty by more than threetimes as much-19 percent. Yet, all transfers go to East Forest residents. Numerous otherexamples of roughly the same approach may be found, with a variety of indicators used.

2'Typically, a Foster, Greer, Thorbecke (1984) index is used, with poverty lines determined using one ofthe methods proposed by Ravallion (1994).

20

Baker and Grosh (1994) examine the implications of geographic targeting in Jamaica andfind that allocating a poverty budget of 10 percent of the poverty line to five (of 14) parisheswould reduce poverty nearly twice as much as a nationwide transfer. Ravallion and Chao(1989) use data on differences in household income across six regions of Java, Indonesia,to show that the optimally targeted budget of 114 million rupiahs would direct the entirepoverty alleviation budget to rural residents of East Java. More strikingly, if one allowstaxation of residents in one region to finance transfers to those in another, residents of EastJava should receive 1,963 Rupiahs per household, while urban residents of West Java shouldpay a head tax of 2,585 Rp. 27

Use of this general approach is not confined to journal articles. In the present inter-national development climate, which features both enhanced efforts to reduce poverty and(at least perceived) tighter fiscal constraints, targeting is very much on economic advis-ers' agenda. A recent treatment of public spending and the poor states that "the mostcommonly heard proposal for achieving a more pro-poor benefit distribution is 'improvedtargeting'." (van de Walle [1995]). A primary aim of household income surveys in LDCs isto construct poverty profiles which are to be used to target resources more effectively. Onerecent World Bank report states that "probably the most important use of the poverty pro-file is to support efforts to target development expenditures towards poorer areas, aimingto reduce aggregate poverty." Policy discussions and most of the articles cited above alikemention political issues in passing, but these are not taken into account explicitly.

5.2 Solving the wrong problem is not right

We consider two versions of the elections approach, in each case using the median votertheorem to pin down the level of taxation at ir(k). In the first version, which reflects themethodology of the targeting assessment literature discussed above, the policymaker takesthe level of taxation as given, while in the second version the policymaker takes the functionr-(k) as given. That is, in the second version the policymaker recognizes that her choice ofk wiU affect the level of taxation in political equilibrium.

We establish a series of results. First, we show that choosing the degree of targeting tomaximize a SWF28 subject to a fixed budget constraint is an ill-posed problem, in the sensethat it almost never produces a 'true" optimum. That is, the budget is not in fact fixed,so that treating it as such involves misspecifying the opportunity set. Second, we offerconditions under which the median voter's optimal tax function r*(k) is monotonicallydecreasing in the degree of targeting. Third, we show in the second version that underbroader conditions, r' is decreasing at k = 0, so that depending on the SWF of choice, itoften will be possible that zero targeting is a (local29 ) Nash equilibrium.

Recalling the social welfare function S and its associated evaluation function G intro-duced in Section 3, we can write the policymaker's problem in Versions I and II, respectively,as:

27Recall Akerlof's result that having some low-skill workers pay taxes to fund transfers to other similarworkers cannot in general be ruled out of the set of optimal policies.

2"Recall that minimizing a poverty index of the FGT form is a special case of this procedure."9We define our concept of local Nash equilibrium below

21

max S(k,r) given r (17)kE[0,1]

max S(k,r)such thatr = r*(k) =: max S(k,T*(k)) (18)kE[0,11 E01

Thus in Version 1, Nash equilibrium requires both that r = r'(k) and that k solve( 17), while in Version II, it is taken as given (by the policymaker) that r = r*(k), soNash equilibrium is achieved by any k solving ( 18). Our first result, given in Theorem 4,demonstrates that Nash equilibria exist in both versions of the Nash approach.

Theorem 4 (Existence of Nash Equilibrium) In both Version I and Version 11, atleast one Nash equilibrium ezists.

Proof: In both of Problems 17 and 18, S is continuous by our assumption that G is, r- iscontinuous by the theorem of the maximum, and the set [0,11 is compact. Hence an optimalvalue of k for the policymaker exists, establishing proof for Version II. In Version I, sinceS also is concave in k, the optimal value is unique; in this case, we have a functional rela-tionship k = 4S(r) if k solves ( 17), where 4 is continuous by the theorem of the maximum.As we describe in the text, the equilibrium set consists of all points (k,r) = (,(r),r-(k)),suggesting we define a function n: F -* F such that n(k, r) = (X(r), r-(k)); note that n iscontinuous and F is convex. By Brouwer's Fixed Point Theorem, n has at least one fixedpoint, proving that a Nash equilibrium exists in Version I.Q.E.D.

Assuming that G-and hence S-is differentiable 30, we can use the usual first order con-ditions to establish necessary conditions for Nash equilibrium. In Version I, each objectivefunction is strictly concave in its choice variable, so (interior) Nash equilibria are fullydescribed by 31

r = T(k), Sk =. (19)

As the discussion in the introductory part of this section details, this situation conformsto what proponents of indicator targeting have in mind. However, our approach in VersionII suggests that more careful analysis is in order. So long as r*(k) E (0, 1), i.e. themiddle class's problem always has an interior solution, our assumptions on u imply, via the

3 0 Some evaluation functions are only piecewise differentiable; one example is the Foster-Greer-Thorbeckepoverty index mentioned earlier. These functions are more tedious to work with, since one must take intoaccount the possibility (perhaps even the likelihood) that optima occur at kink points. However, the usualapproach, i.e. checking all kink points for optimality conditions, solves the problem; as there is no gain inintuition from treating such functions explicitly, we leave the reader to make the necessary modifications tothe argument in the text.

3'For boundary solutions, we would replace = with < (for k = 0) or > (for k = 1). Of course, Theorem Idemonstrates that at k = 0 we have Si > 0.

22

implicit function theorem, that r* is everywhere differentiable. Hence we can take a first-order approach to ( 18), so that the necessary condition for (an interior) Nash equilibriumof Version II is32

Sk + ST dT = ° (20)dk

Except for points where either S, or dr*/dk equals zero, optima will not be characterizedby Sk = 0. This fact of course begs the question, Will ST or dr-/dk equal zero? Beforeanswering, we demonstrate in the following theorem that from the policymaker's point ofview, any Version 11 equilibrium weakly dominates all Version I equilibria. The logic ofthe theorem is nearly trivial: if one misspecifies one's constraint set and then solves theassociated misspecified optimization problem, but the constraint must bind in equilibrium,one can hardly expect to outperform an optimal choice made over the correctly specifiedconstraint set. The "weak" part of the theorem is necessary because it is of course possiblethat the misspecified constraint set contains the optimal choice from the correctly specifiedone. However, after stating the theorem, we shall argue for the unlikeliness of this particularoutcome in the present context.

Theorem 5 (Assuming a fixed budget is (almost) never optimal) Suppose G is dif-ferentiable. Let 4 be the parameter space with vector elements 4' satisfying the basic re-strictions of the model. Let

VI(O) = {k: k solves ( 17) when the parameters of the model are specified as 4d>

and define VII(*) analogously. Then for any K E VII(*) and r.' E VI(4'), S(Kc,Tr(Kc)) >S(KC', r*(x')).

Proof: Suppose that K1 solves (17); then by definition Sk(oc', r(Wc)) = 0; note that byTheorem 1, ic' > 0. Hence from ( 18) we have

dS I('~K d)~Kr* KI (1Jk=S,(c'r. (' dk (Wc,r'(Wc)) (21)

So long as neither term on the RHS of Equation 21 is zero, it is obvious that K1 doesnot solve ( 18), which is to say that there is some other value r. in a neighborhood of /c'such that S(cr,T(K)) > S(C', e(o')), If either RHS term is zero at K1, this inequality neednot hold (it need not be violated either, since it is possible that K' is a global maximum ofS over the Version I constraint set but only a local maximum over the Version II), but K'

itself is always feasible in ( 18), so the policymaker never does worse by solving the VersionII problem.

Q.E.D.

32A above, we note that boundary solutions would involve the replacement of = with < (for k = 0) or >(for k = 1)2

23

Figure 4a should provide the necessary intuition. The curves labeled SW are iso-social welfare loci, whose slopes are -Sk/Sr. At point A, or (ko,ro) the curve SW0 hasa minimum, so Sk = 0; also, r*(kO) = ro. Hence A is a Nash equilibrium of Version I.However, dr*/dk > 0 at A, so that an improvement of social welfare is feasible by moving,for example, to point B: Version I leads to an undertargeted equilibrium in this case. Thisoutcome occurs because when the policymaker takes to as given, she ignores the entireregion R, all of which is feasible and some of which allows welfare improvement over A.

Figure 4b shows an example where the equilibria of our two games are equivalent fromthe policymaker's perspective. It should be obvious that the possibility of this event isquite remote 33.

5.3 Is the "naive" amount of targeting too high or too low?

Although Figure 4a showed an undertargeting example, we believe that the more likelyscenario is depicted in Figure 5. The difference here is that at A', dr*fdk < 0. In this case,the policymaker fails to regard as feasible the region R', all of which allows improvement. Inthis case, the Nash equilibrium of Version II occurs at (0,r*(O)), where the policymaker ison her highest politically feasible iso-welfare curve. At this point, the 'naive" policymakerof Version I believes the entire rectangle r*(O) - 0 - I - r*(O) is feasible, so she turns downthe opportunity to select (O,r*(O)), preferring instead the point C, with the associated levelof targeting kl. But at k1, rs(O) is not selected by voters; rather, they choose ri, and soon, until we reach the point A', where both Sk = 0 and r = r-(k).

Figure 5 suggests heuristically that when the Version I equilibrium is worse than theVersion II equilbrium, it can be a lot worse. This fact suggests that we should examine indetail the conditions under which our economy looks like the one depicted in Figure 2. Inparticular, we would like to characterize dr*/dk and the sign of S,, especially at k = 0. Wewili find it convenient to restrict consideration to the case of constant relative risk aversionutilities 34.

Unless we indicate otherwise, we assume throughout the remainder of this section thatC 1 is satisfied with strict inequality. We proceed by offering sufficient conditions underwhich rT decreases monotonically. We then show that at k = 0 this sufficient condition isfar stronger than necessary.

Proposition 1 Suppose that we have CRRA utilities with parameter p, C I is satisfiedstrictly and p < 1. Then r is monotonically decreasing.

Proof: To begin, we recall from equation ( 13) that dr/dk = -(82 U./,kkr)/(8 2 U../Lgr2 ).Since O2 Umn/r 2 always is negative (by concavity of u), dr*/dk has the same sign as892 Un/Mkor. Dropping the m subscript of Um, partial differentiation with respect to kyields

33 Algebraically, this event requires that there is a point (k, r) where simultaneously MUm8tr =8 2 UmU8kt9r = Sg. = 0. There is no reason why all three of these conditions-which generally imply verydifferent qualitative restrictions on the model's parameters-should hold at once.

34When we say that this restriction is for convenience, we really mean it. The main result we report inthe text below carries through without the CRRA assumption; we will use footnotes in the text below todescribe the necessary revisions when a general utility function is used.

24

I

K-k

F%m 4b:

Fipiw '.

MAn I ex twi bichteamw" puva -

c 100 linic . . _.

A. poL;_Eiihbpes to the

aceamy opmak lee of

6mTy{r([pU'(N + 8) + qu'(N + 0 + 1 - r)] - ym[pu'(N) + qu'(N + 1 - r)J} (22)

Differentiating this expression partially with respect to r, we have

UkT = -+ 6m.Y{;I[p(N + 0)u"(N + 0) + q(N + 0 - r)u"(N + 0 + I -r)

- 7 m[pNu"(N) + q(N + 1 - r)u"(N + 1 -r)]}, (23)

which can be rewritten as

Uk 7 =

m.Y(I - p){4[pu'(N + 0) + qu'(N + 0 + I - ]-Ym[Pu'(N) + qu'(N + 1 -r)]}

-q6mY[;Tu'(N + 0 + 1 - r) -tmu"(N + 1 - r)] (24)

Since u" < 0 and 7y < ym (by C 1), the term multiplied by (1 - p) is negative. The restof the RHS must be negative, since u"' > 0. lience p < 1 is a sufficient condition for r' tobe monotonically decreasing.Q.E.D.

That rT should be monotonically decreasing whenever the degree of risk aversion doesnot exceed that exhibited by log utilities has an entirely intuitive explanation. 35 From thepoint of view of middle class voters, the role served by taxation is the redistribution ofincome from the unemployed state of the world to the employed one. When C 1 is satisfied,we know from Lemma 1 that this redistribution becomes less efficient.36 Stated in slightlydifferent form, as k rises, the price of shifting consumption from the employed state tothe unemployed state rises. We therefore know there wiU be a substitution effect: when agood's price rises, people have an incentive to shift some consumption to other goods. Inthe present case, "other goods" are employed-state consumption, so that the substitution

35We can weaken Proposition l's hypothesis by dropping the CRRA assumption and instead substitutingthe assumptions that i.". > 0, p be nondecreasing, and p(l) C 1. The assumption on u."' ensures negativityof the second part of the RHS of equation ( 24). As for the first term, it must be altered by multiplyingeach u' term by one minus its associated coefficient of relative risk aversion (e.g. we multiply u'(N) by(I - p(Y)]. We require p(l) S 1 because V < q < I gurantees N < r, so that N < N + I - r < 1. Togetherwith the assumption that p is nondecreasing, this ensures that neither term multiplied by 7ym exceeds itsassociated 3 term, It is worth noting that while the degree of risk aversion is uniquely determined (unlikethe representation of preferences over consumption levels, the representation of preferences over risk is notaltered by affine transformations), our requirement on p(l) might seem to imply that when u does notsatisfy the CRRA form, it matters which units we use to measure income. Of course, this is backwards: ifwe multiply all income variables by any positive constant, middle class voters' first order condition U,. = 0is unaffected, and we end up with the condition that p(m) < 1, where m is middle class voters' maximummarginal product. Hence the restrictions in this footnote are qualitative in nature.

3"This point is made precise by observing that 92 Um/okar = i9(aUm/ar)/ak, so that the conditiona2Uml/9k49r < 0 implies that the change in utility achieved for a smal increase in the tax rate-OUm/a7r-isdecreasing as k increases. It is therefore natural to think of increases in k as reducing the efficiency oftaxation in redistributing income across states of the world.

25

iwund Anwim iihmIn nz i mm.DAO .~m Atodwu

4~~~~~~~~~~ Iu~ DI aT>

I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

I w Ee Sn -..

/ I

ts} .. o*1

effect makes middle-class agents want to reduce the tax rate. Of course, there will also bean income effect: since any given level of consumption of a good becomes more expensiveas that good's price rises, more income must be spent on that good to maintain the presentlevel of consumption. Thus the income effect makes middle-class agents want to increasethe tax rate.

Hence the signing of dr'/dk can be thought of as the resolution of the standard tensionbetween the substitution and income effects. When p is unity, we have the Cobb-Douglas(log utilities) case, when we know that the income and substitution effects exactly offset;thus we could hardly expect 7` to be increasing in this case. While it might seem puzzlingthat p = 1 is a sufficient condition for r* to be decreasing, since we have just noted theoffsetting influences of the substitution and income effects in the Cobb-Douglas case, thepuzzle is solved when we recognize that all prices change when k changes. That is, changesin k change the relative rate of redistribution of income across states in a non-proportionalway, so that more than one set of income and substitution effects must be taken intoaccount.

It is reasonable to believe that people's risk aversion exceeds that described by log util-ities, and we admit both our desire and our inability to derive general analytical conditionsthat allow the signing of dr*/dk in all cases. Ilowever, it turns out that the log utilitiescondition is much stronger than is often needed. We demonstrate this fact by consideringthe case when k = 0 and showing that under broad circumstances, rT may be decreasingthere even as p approaches infinity. At k = 0, we have 0 = 0 and N = yr, so that the firstorder condition is

p-yN- = q(l - y)(N+ l -) (25)

We can solve this equation for the explict value of r'(0):

I

r-(O) P where7+ (1i - v Ax

PY =(26)q(l -y

It is easy to see that so long as y < q and p > 0, we have A, T(O) E (0,1). It isalso clear that UmProo r (O) = 1 and limP-o r-(O) = 0. These facts confirm what ourbasic intuition should be: with no targeting, the tax rate (and hence redistribution acrossemployment states) should approach zero as people approach risk neutrality and unity as themarginal rate of substitution of income between states becomes infinitely inelastic (Leontiefpreferences). In the special case when p = 1, i.e. log utilities, we have r*(0) = p/(l - y).In general, r(0) is increasing in r,a,m, and r (since the transfer "technology" is moreefficient from the perspective of any given middle class voter the greater is the rich-incomepart of the tax base and, in general, the more other people-including other middle classpeople-there are to tax). The general effect of p is indeterminate, i.e. it depends on p, sincea higher probability of unemployment for all middle class people both increases their need

26

for transfers and decreases the tax base, making inter-employment state transfers moreexpensive for the middle class37.

Next, recalling that dir/dk has the same sign as 02Um/OkTr, we take this cross-partialderivative and divide it by tmY, yielding

(Y -m)I[pu'(N)(l - p) + qu'(N + 1 - r)(1 - p) - qu"(N + 1 -)]

The term in parentheses has the same sign as bm - 6, which we have assumed to benegative. Noting that u"(c) = -pu'(c)/c and that our first order condition for r*(O) waspyu'(N) = q(l - y)u'(N + 1 - r), the term in brackets will be positive if and only if

p < I + -yj + p N (27)

Using Equation ( 26) it can be shown that (r - N)/(l - (r - N)) = A1/P(1 - Y)/Y, sowe can rearrange ( 27) to read

1 + .p < - + PAP

or

p(l - AP) < l _(28)

The right hand side of ( 28) is strictly greater than unity, and the left hand side isstrictly less than p, leaving a good deal of slack (as theory goes) for satisfaction of ( 28); inparticular, if p < 1-i.e. the degree of risk aversion is no greater than that present with logutilities-then ( 28) must be satisfied. To consider p > 1, we note that the left hand side of( 28) approaches zero when p approaches zero and is strictly concave in p. By L'Hopital'srule, hm,_OO LHS = - In A > 0, so the left hand side must also be strictly increasing;hence - In A is the least upper bound of the LHS. Therefore, a sufficient condition for thebracketed term in equation ( 27) to be positive, no matter what value p takes, is

-InA <

or

A > exp[- j--] (29)

Evaluating at y = q, the upper bound on y the left hand side of ( 29) is unity and theright hand side is exp[-l/(1-q)]; if q = 0.9, then this term is exp[-10], which is 4.5x 10'.Quite obviously, y may be very small and still satisfy the inequality. To be more concretestill, suppose that q = 0.9 (there is a ten percent chance of unemployment), arr = 0.6, and

3 7 0f course, if we carry out the thought experiment of raising a given middle class person's probabilityof unemployment while holding everyone else's (and hence the tax base) fixed, it is of course the case thatthat person will want a greater tax rate.

27

am = 0.3. Then g = .87, the right hand side of ( 29) equals 0.00046, and A = 0.088; wesatisfy our extreme sufficient condition with two orders of magnitude and almost a factorof two to spare. If one enjoys plugging parameters into models, one could hardly seek astronger confirmation of a claim than is provided for our argument that dr-/dk < 0 atk = 0 when bm < 6

Continuing with our analytical treatment of the Version II model, we note that ingeneral, one would expect that if the policymaker is interested in poverty reduction, wewill almost always have S, > 0 (unless r is very large), i.e. from the policymaker's pointof view a slightly greater value of r would be an improvement. This assertion is buttressedby the fact that in any Nash equilibrium, r = r-(k), so that (if T E (0,1)) 9Um/&r = 0.If we define S' as that part of S that corresponds to group i's utility, i.e. S'(k,T) =EhE rhju(yi), then in any Nash equilibrium we will have S' > 0, with equality onlyif S is utilitarian (i.e. G is the identity function). lhence we will always have S7 > 0unless ojS' < -aTS, i.e. unless the policymaker views further taxes as detracting more(according to the metric implicit in G) from the rich than adding to the poor. When there isno targeting, it is straightforward to show that this event is impossible. We show the resultwhen S is utilitarian; adding more curvature (i.e. making G a strictly concave function)only strengthens the result. Since we are evaluating at k = 0,

S. = y[pu'(N) + qu'(N + JL)]

and

Sr= -(r - V)u'(N + r[l - r])

Since rich agents' incentive compatibility constraint must be satisfied, we have r([ - T] <p. Hence the u' terms are greater for poor than for rich agents. As such, a sufficientcondition for S,(O, T) to be positive is ogy > o,,(r - y), which can be shown to be equivalentto the requirement that y < q, which has already been assumed. Ilence S(O, r) > 0 for allr satisfying (IC)r.

If S, is positive and dr*/dk is negative, then Equation ( 20), which can be rewrittenas 5 = -S,drl/dk, implies that at any optimum, Sk must be positive! Under theseconditions, when empirical work (like that discussed in the introduction to this section)shows that, fixing the budget, an increase in targeting will improve social welfare, we arenot entitled to conclude that transfers are being delivered inefficiently-at least not withoutprior knowledge that the conditions for Sr > 0 > dr'/dk are violated.

This result really is the heart of our criticism of policy recommendations based onempirical findings that Sk > 0, i.e. that the implementation of indicator targeting wouldreduce poverty given a fixed rate of taxation. The approach is incapable of distinguishingwhether the explanation of the apparent gain from increased targeting is inefficient policydesign or a combination of real concern for efficiency in targeting the poor tempered by anawareness of the endogeneity of the budget with respect to increased targeting.

28

5.4 When is zero targeting a Nash equilibrium?

We need not stop there. We now offer an example in which no targeting is a (local) optimum,and hence a (local) Nash equilibrium.38

With CRRA utilities, it can be shown that at k = 0 we have

-(I - 4 )T*(O)Z, (30)ly 6

where

z -1+ Py+ ( )P

We note that z > I if and only p < 1.Now suppose that the policymaker is as pro-poor as possible, and chooses as her social

welfare function the utility of a representative poor person. Then -SkIS 7 = dri/dk. Fromthe proof of Lemma 3 it can be shown that at the point (O,r*(O)), we have

dkrl( T(01r )) = (- - 0)T*(°) (31)dTk

Hence zero targeted transfers is (locally) optimal under Version II if and only if the RHSof ( 31) exceeds the expression in ( 30), i.e. if and only if the politically-induced reductionin the tax rate that occurs when a small amount of targeting is introduced exceeds thereduction in the tax rate that would leave the poor just indifferent given that amount oftargeting. To think of the situation graphically, simply replace the SW curves of Figure 5with the indifference curves TI: zero targeted transfers is locally optimal if and only r(k)lies outside poor agents' better-than set for all k sufficiently close to zero. Algebraically,we have optimality if and only if

d 61 < Z( 6 -mJ L _ ( 32)

This condition may be rewritten as follows:

- < I -a 1 7 (33)

'The concept of a "local" Nash equilibrium is not standard; what we mean in that there exists someopen neighborhood of the point (O,r'(0)) such that if the game were restricted to that neighborhood,(0, r'(0)) would be a Nash equilibrium of the restricted game. As we discussed above, since in Version 11the policymaker takes r'(k) -which is the politico-equilibrium value of r conditional on k-as given, Nashequilibrium is fully described by any choice of k solving ( 18), a necessary condition for which is that k solveequation ( 20). Unfortunately, no primitive conditions exist under which the function V(k) S(k, r (k))is globally concave in k, so there is no guarantee that any particular solution of equation ( 20) is unique.Moreover, there remains the question of the second order sufficiency condition. Fortunately, since we aredealing with a boundary point, our argument will seek to establish conditions under which S& +S?dr7ldk isstrictly less than zero when all terms are evaluated at (0, r'(0)). If this inequality holds, then we must havea local maximum by definition, since: 1) a local maximum is any point k where V(k') < V(k) for all feasiblek' in some neighborhood of k, 2) V'(0) < 0 implies there is some c such that for all k' E (0, r), V(O) > V(k')and 3) k' < 0 is not feasible.

29

This condition requires a given middle class person to be sufficiently less likely thana given poor person to receive the targeted transfer. Recall that in condition C 2 ofsubsection 4.2, however, we required the opposite to hold: poor and middle class agentshad to have sufficiently close probabilities of receiving targeted transfers (the condition wasthat 6m/6l 2 1 - q/r). Thus it is interesting to know when the two conditions may besatisfied jointly and when they are mutually exclusive; they may be satisfied jointly if andonly if

q > ar y+1 r - (I1- a.)Vz -am(1-

This inequality holds if and only if z > z (=1 - y)/(q - y). It is clear that z' is strictlygreater than unity, so that p must be less than unity if our results are to be nonexclusive.While we are disappointed with this finding, p need not be too much less than unity; it canbe shown that a sufficient condition for z > z* is for p to be less than

P* q Y [X+q

As an example, with p = 0.1 and y = 0.6, we would need p < 0.82, not so different fromlog utilities. It is important to recall that when p = p*, there is but a single value of theratio Am/l6 for which both our results may hold. While this fact implies much lower levelsof risk aversion for which one might feel comfortable saying that a broad class of parametervalues are likely to satisfy our conditions, it is still the case that at least one of our twoapproaches will suggest that targeting fails the political test. Put another way, the level ofrisk aversion will have to be fairly high before one might comfortably assert that neither ofour critiques might be valid for a broad class of parameter values.

Moreover, we stress that the low level of p required for both results to hold is an artefactof our choice of social welfare function (the utility of a representative poor agent). Instead,we might have chosen, say, a convex combination of all three agents' utilities.3 9 In fact,weightings always exist for which a choice of zero targeting is optimal for the policymaker,no matter what level risk aversion takes.40 These facts go a long way in cementing ourbelief that both the social choice and Nash equilibrium approaches are useful in assessingthe viability of indicator targeting.

It might be objected that the whole project of assuming a social welfare maximizingpolicymaker is a sham, anyway. But this argument is a nonstarter within the contexts ofcurrent debate over targeting, since the claim being advanced by proponents of targetingis precisely that social welfare would be improved with more targeting. Moreover, it seemsreasonable to believe that policymakers typically are at least as concerned by politicalviability as they are about first-best (or n - 1 best) program design. Also, policymakers arelikely to hold a comparative advantage over economists in assessing the political viabilityof various policies.

39Such a choice unambiguously would restrict the social better-than set. Put another way, the slope-ShI/S, is increasing in the amount of weight put on middle clasn or rich agents' utility.

4"That is, there will always be some weighting under which our condition for local (politically constrained)optimality of zero targeting requires only that 6m,/6i not exceed a number greater than unity; but 6,,, < 61by assumption.

30

To conclude this section, we do not claim that whenever non-targeted programs areobserved, it must be the case that the status quo is optimal, even taking political factorsinto account. However, we think that the argument advanced in this section goes a longway in shifting the burden of proof toward proponents of targeting: it is not enough simplyto declare that in a world of fixed budgets, program efficiency can be enhanced by moretargeting. Rather, the political costs of more focused transfer delivery must be researched,evaluated, and weighed against the economic benefits.

6 Empirical Facts Consistent with our Model

In this section, we argue that our model has direct practical relevance, particularly topolicies actively being considered in LDCs. We first consider policy prescriptions flowingfrom recent empirical work and then turn to some empirical evidence that both is at oddswith those prescriptions and supports our model. While we are not silly enough to thinkthat a model as schematic and stylized as ours can be "tested" in any rigorous sense, wedo think our model is broadly consistent with three strands of evidence about governmentspending and targeting.

First, nearly every incidence study of benefits of government expenditures-even thoseaimed at social-welfare, e.g. health and education-finds that benefits are at best no moreprogressive than would be a uniform transfer to all citizens. While social expenditures attimes are progressive relative to pre-transfer income distribution, typically they are muchless progessively distributed than would be a uniform transfer. Table 3 summarizes theresults of a number of such studies, whose subjects include social insurance, health, andeducation. These programs typically constitute the bulk of social spending in developingcountries. We report the ratio of benefits received by the third ("middle class") and fifth("richest") quntile to those received by the poorest quintile. In nearly every instance,the per capita benefits received by the middle class and the rich exceed those receivedby the poor.4". Even in programs justified exdusively on the grounds that they aid thepoor, the incidence of benefits is less progressive than would be a uniform transfer (see, forinstance, Alderman and von Braun (1984) on Egypt's food subsidy). As Birdsall and James(1993) point out, a normative model of how an egalitarian social welfare maximizer woulddistribute benefits makes a poor positive model of the actual distribution of government-funded benefits.

The second fact supporting our conclusions is that episodes of increased targeting havebeen followed by reductions in overall benefits, as we would predict. In the mid 1970sSri Lanka had universal food subsidies that provided every citizen with a rice ration atsubsidized prices. At around 5 percent of GDP, the program's fiscal cost was felt to beunaffordable. In 1978 the government restricted the ration to the poorest half of the

410f course the distribution of benefits within the social spending categories is very different (e.g. uni-versity versus primary schooling, public health versus hospitals). For instance, Jimenez (1994) finds thatthe median ratio of the fraction of benefits received by the lower 40 percent of households to the middle 40percent for all education spending is 1.23 (for eight countries) and of public health (not public spending onhealth) is 1.27. Moreover, since poorer households tend to be larger than other ones, this fraction will begreater than the analogous one using per capita benefits.

31

Table 3: The ratio of benefits per person from various types of social expenditures accruing toindividuals in the third (middle) or fifth (richest) group' relative to the poorest.

Country Type of social expenditure Ratio of benefits b income group:

Third to poorest Richest topoorest

Brazil Education 1.18 1.55

Health 1.73 1.01

Social Insurance 6.31 12.34_

Total Social Spending (including 1.81 2.31other categories not shown) ._l

Vietnam Education 1.48 3.38|

Health 1.80 2.49

Social Transfers 2.62 5.15

Uruguay Education 0.51 0.44

Health 0.47 0.29

Social Security 1.65 2.48

Total Social Spending (including 0.91 1.16other categories not shown)

Tanzania Education (Primary and 1.14 2.64Secondary)

Health .94 1.70

Total (including water) 1.14 2.42

Ecuador Education 1.32 1.43

Health 0.98 1.14

Kenya Education (Primary and 1.18 0.97Secondary)

Health 1.57 2.33

Peru Education 1.67 1.67

Colombia Health 1.06 0.79

Rural Programs 0.81 0.51

Indonesia Health 1.56 2.42

1) The classifying groups are quintiles of either per capita consumption or per capita incomeexcept in Brazil, where the fives classes are defined by household income relative to theminimum wage (e.g. < 1/4 of minimum wage, 1/4 to 1/2, 1/2 to 1. I to 2, >2).Sources: For Indonesia, van de Walle, 1994, for Peru, Selden and Wasylenko, 1995. for therest various World Bank reports.

population and in 1979 replaced the rice ration with a food stamp program, again targetedto the bottom half of the population. Since the introduction of targeting the real value ofbenefits per household has fallen to 40 percent of its previous level, from Rs 425 to Rs 170.It is intriguing to note that even with targeting there was substantial leakage. Calculationsfrom a household survey indicate that (in our notation) 6,m = .38 < 6 = 0.43 < bl = .7.These parameters fit our story, and it is also the case that targeting was far from "perfect",so that the non-poor had ample chance in absolute terms to receive the targeted subsidy.While it is impossible to say what would have happened to the program's budget hadbenefits remained universal (since 5 percent of GDP simply may have been unsustainable),the fiscal cost of the program is now just 0.7 percent of GDP. More of the savings camefrom reductions in the magnitude of the transfer than from better targeting.

A similar episode ocurred in Colombia, where a targeted food stamp program waseliminated. In terms of the differences between the technocratic and political approaches, arecent assessment of poverty in Colombia (World Bank, [1994]) seems instructive: "althoughthe program seemed effective and well targeted in Colombia, it lacked political support andwas discontinued." Our results suggest that the word 'although" should be changed to"because".

Notwithstanding our earlier disclaimer regarding our model's lack of direct applicabilityto developed countries, trends in U.S. social welfare spending constitute another potentialexample. Only in the most radicaly of anti-government climates has Medicare become a po-litically viable target, and Social Security remains totally off-limits to budget cutters. Bothof these programs, while benefiting only the elderly at any given moment, are received uni-versally (at least if the labor force is considered universal) by people reaching retirement age.By contrast, every imaginable "welfare" program-AFDC, food stamps, Medicaid-targetedto the poor has rested squarely on the anti-government chopping block.42 Interestingly,unemployment benefits, which clearly do benefit the working and middle classes, have beenmaintained in real terms. In fact, during recessions, when the proportion of potential andactual middle class UI recipients is comparatively large, benefit duration often is extended.Moreover, we are not the only analysts to suggest a positive link between the degree oftargeting for such programs and the erosion of political support for their financing (see, forexample, Skocpol [1991]).

Our third strand of supportive evidence is the cross-country relationship between themagnitude of social welfare spending and the degree to which it is targeted. Figure 6 (takenfrom Milanovic [1994]) shows the association between the degree to which cash transfersare targeted and their overall magnitude (as a fraction of GDP). The U.S. finds itself at onepole, with highly targeted but small overall transfers, while pre-market-transition EasternEuropean countries inhabit the other extreme-large transfers with hardly any targeting.

421t will be interesting to see what happens to Medicaid funding given the exploding share of programspending now directed at nursing home costs for those elderly people who qualify for Medicaid only afterexhausting (often-substantial) savings on such care.

32

Iigmie 6: Itevladioisliip bel[veen the magnitude of cash transrers and their progressivity

10 BUL .

H1eN *S0

-10 t

-20 UK; I

(i8A

1 30 )UCAP' ,AGER -

-40 NOR

0 a 10 1l 20 25 30

Transrcrs as a percent of bousehold gross income

S;auircg: MIilauw"ic '199J

7 Altruism

In this section, we generalize the model to include a treatment of altruism and then considerempirical findings concerning the extent of altruism. We believe these findings are incon-sistent with the hypothesis that altruism can be an important factor in generating publiclyfinanced income redistribution. Nonetheless, we use evidence from Sri Lanka to construct aback-of-the-envelope calculation of the degree of altruism necessary to invalidate our claimthat the general, altruism-included model can be invoked to explain experiences (like SriLanka's) with targeting.

7.1 Incorporating Altruism Into The Model

We assume that the "overall" utility of the middle class and the rich may be writtenas a convex combination of own utility and the expected utility of a representative poorperson.43 This approach has the virtue of being easy to think about intuitively: if a is theweight a person places on her own consumption, then we might think of the given personas admitting a probability distribution over her type, with the probability that she is tobe of her actual type equaling a and the probability that she is to be poor equaling 1 - a.Hence the approach has a certain Rawisian flavor: people can be thought of as consideringthe possibility that they might have been in someone else's shoes before making decisionsabout social policy.44

We define the functions

V,m(k,r) _GUm(k,T) + (1 - a)Ul(k,r)

and

V7. (k, r) _3Ur(k, r) + (1 - i3)U1(k, r)

so that a and , are the respective weights placed on own utility by middle class andrich agents (we assume that the poor care only about their own consumption and thatthe rich do not care about middle class consumption). It will be useful to define the

'3 The convex combination part of this assumption has no substance, as it follows directly from theusumption of additive separability: suppose Vm _ oU, + $1Ur for arbitrary non-negative constants n, ,3;then all choices regarding targeting and taxation are the same as if VI/(ar + d) is used, since this secondfunction is a monotonic transformation of the first, and the new weights sum to unity.

"It is worth flagging a basic conceptual problem with this approach: if there are positive levels of targetedtransfers, some people who are poor by type, i.e. have maximum marginal product A, will have realizedpost -transfer income exceeding that of some middle class type agents. This event occurs because somepoor people receive the targeted transfer 0, while some middle class people are both unemployed and do notreceive 9. Moreover, if taxes and targeted transfers are large relative to universal transfers and the marginalproducts of middle class or rich agents, poor people receiving targeted transfers may receive post-transferincome exceeding that of even employed middle class or rich agents. Hence the convex combination approachcarries with it an explicitly ex ante character: it is best interpreted as altruism defined over possibilities, notoutcomes. One might try to remedy this problem by placing a ceiling on income levels subject to altruisticfeelings, but we reject this approach, since it would only strengthen our claims while playing havoc with themodel's tractability.

33

function r**(k) _ arg maxt Vm(k, t). As with r', elementary mathematical results ensurethe existence, continuity, and, when aVm(k, 1)/Tr < 0, differentiability of r*T.

To review the basic steps taken in deriving our primary results above, we used thefollowing facts, true under assumptions C 1 and C 2:45

(I) t9Um/Ok < 0, with equality possible only at k = 0(II) OUl/ak < 0(III) Single-crossing between any two intersecting Ti- and T- curves, with the former

crossing the latter from below(IV) a2Um/a9kOr < 0 for CRRA utilities with p < 1To show (I), we offered a descriptive proof, using the fact that when 6m < 6, i.e. when

C 1 holds, targeted transfers are inferior in expected value to an actuarially fair gamble; theresult followed directly from this fact. For the general case, i.e. without making assumptionson u or the particular proportion 6 of people receiving targeted transfers, this approach isthe only way to obtain the result. A similar condition can be derived when we admit thepossibility of altruism, but it is less general.

At first glance, the logic of our proof of Lemma 1 suggests that we need only continuewith our Rawisian interpretation of altruism, and derive a condition under which the total"probability" that our non-poor agent will receive a targeted transfer, taking into accountthe "probability" that she will be poor", is less than the proportion of the populationreceiving the transfer. IHowever, it turns out that if we want our result to hold for generalpreferences, this condition, which may be written as acrm + (I - a)6, < 6 for the middleclass,46 cannot always be shown to be sufficient. The basic logic is as follows. If 0+p < l-r,employed middle class people-even those not receiving 0-always have higher consumptionthan do employed poor people. Hence with probability 1 - a, employed middle class peoplewho "become" employed poor people will "need" 0 more than will employed middle classpeople who 'remain" middle class.47 As such, these middle class people will place a premiumon targeted transfers, reflecting the chance that poor people whose need for them is greaterwill receive them. Hence without saying more about the shape of u, we are entitled to usethe above condition only when O+p > I - r, in which case the opposite calculation is made:employed middle class people not receiving 0 have a higher marginal utility of consumptionthan do employed poor people who do receive 0. For the rich, however, the requirementon 0 is stronger still: we must have 0 + p > r(l - r); all other details of the precedingdiscussion apply directly.

45We wish to note that in this section we do not deal with a number of technical issues handled in theno-altruism case (e.g. accounting for the incentive compatibility constraint when p 4 0). While a fuUltreatment would require some such issues to be resolved, others of them no longer are important, largelybecause the introduction of altruism requires that all attention in the social choice approach be focused onlocal rather than global equilibria. We have chosen to ignore those details that remain, on the grounds thatthere is nothing intuitive or challenging in handling them.

"For the rich, since 6, = 0, the condition would be (I - ,)6b < 6, but this condition will turn out to holdin all cases we consider below.

47Our focus is on the employed of both classes because in the event of unemployment, members of bothclasses receive zero pre-transfer income. Hence differences in post-transfer income for unemployed agentsderive solely from differences in 6b and 6,. lhence the 'naive" condition in the text is sufficient for that partof expected utility related to the unemployed state, i.e. that part multiplied by p.

34

We now offer two very similar conditions that allow us to generate in Lemma 5 resultsanalogous to (I) and (IV).

C 3

cram + (1 - a)bl + q(1 - ca)(6L - 6)(u'(A) - 1) < 6

C 4

o4,, + (1 )6l + (1- )(<, -3 )(u'(s)-1) < 6

Lemma 5 (When Altruistic Middle Class Agents Oppose Targeting) (i) SupposeC 3 is satisfied. Then 49Vm/t9k < 0 for k = 0.

(ii) Suppose that C 4 is satisfied and that utilities are of the CRRA form, with p < 1.Then 022Vm/OkOr < 0 for k = 0.

Proof: See Appendix 3.

The intuitive explanation of why C 3 gives the lemma's first result is simple enough. Theterm Ga6m + (1 - a)6i plays the role explained above, while the remaining term on the LHSis a correction factor for the greater marginal utility of income experienced by employedpoor than employed middle class people. This correction term has four components. Theq occurs because the correction need be made only for differences in middle class and poormarginal utilities occurring in the employed state. The (1- a) part reflects the fact thatthe problem arises only when middle class agents place some weight on the utility of thepoor. The (6i - T) term is a measure of the gain (relative to an actuarially fair gamble)from targeting received by a poor person (at k = 0). The last term, (u'(%) - 1), reflectsthe difference in marginal utilties of consumption between employed poor and middle-classagents (as we detail in the lemma's proof, we choose I as a normalization for u'(1)).

As for the second condition and the lemma's second result, the only difference in thetwo conditions is the absence of the q in the third LHS term in C 4. Explaining why thisdifference is necessary is less easy than explaining the roles of the other components of thecondition. Because the poor are not taxed while middle class are, an increase in targetinghas a different (i.e. non-proportional) impact on the redistributive efficiency of taxationfor the two types of agents. Beyond that, all that can be said is that the algebra playsout in such a way that an analytical sufficient condition may be obtained only via thestrengthening of C 3 into C 4.

It should be easy to see that the deviation of C 3 from its naive counterpart is increasingin the degree of altruism (i.e. in 1 - a, the weight middle class people put on the utilityof the poor). Second, this deviation is increasing in the excess of 6i over 6, since thepoor benefit from targeting more as they become relatively more likely to receive targetedtransfers. Third, the deviation of C 3 from its naive counterpart is increasing in the excess

35

of the marginal utility of consumption of the employed poor over that of the middle class,since the greater is this difference, the greater is the relative need of the employed poor(by comparison to the employed middle class), and hence the greater is the "premium"middle class agents place on the value of targeting. It is important to note that except forthe income of the rich and the probability of unemployment, all of the model's parametersnow affect our (strong) sufficient condition for middle class agents' utility to be strictlydecreasing in the degree of targeting. Hence more risk aversion or lower (relative) income ofthe poor each makes violation of the condition more likely, just as altruism does. Moreover,these effects interact, so that small amounts of altruism in very poor, very well targeted(i.e. 61 >> ) economies may lead to the violation of the condition; the converse is of coursealso true: large degrees of altruism will make little difference (relative to the basic Rawlsiancondition) in economies with fairly equal left-tail income distributions and poor targetingtechnology.

The second result of the previous lemma is as far as we will go in extending the Nashequilibrium results. The lemma makes it clear that the condition under which dr**/dk < 0-namely that OUml/kCOr < 0-is not knife edge in character. Moreover, we can always choosean a close enough to unity to be in a close enough neighborhood of the no-altruism case.However, to actually calculate a threshold level c* (as a function of the model's otherparameters) such that the Nash equilibrium results of Section 5 hold for a > a would betime consuming, and, we think, not terribly intuitive. The point is that such a level mustexist and, for fixed parameters, be bounded away from unity. By contrast, our social choiceresults may be generalized analytically without much trouble.

For reasons that will become apparent, our results in this section do not require usto offer conditions under which 9VI/Ok < 0; this result will be shown to follow directlyfrom other conditions developed here. As for extending result (III), this task might seemforeboding: it now requires that

drl _ OUI/OkSik OU 01/rdi% P8Ur/Ok + (I -,3)OUI/Jk (34)dk baU 7/13r + (I - 3)OU,i/Or

and at first blush, there seems no particular reason to believe that C 2 should generalizedirectly. Nonetheless, this concern will be shown to be without foundation.

We begin by defining the following relation (which may be either a function or corre-spondence) analogous to rr:

;,(k; V) -{r: Vr(k, r) = V}

Hence the family of f, maps describes the level sets of now-altruistic rich agents. Beforeproving two useful lemmas, we note that the equilibrium status of (0,r**(0)) to be shownbelow in general is neither global nor local: we show in Theorem 6 that, under certainconditions, there is a definite, non-local part of the feasible set over which this point can beshown to have equilibrium status, even though that part need not be the whole set. While

36

the result to be proved is a single-crossing result, there may be multiple segments to thelevel set ;,, so that even when single-crossing holds for that part of f,. "near" (O,r**(O)),the point (0, T*(O)) is not a global equilibrium. Such a situation is depicted in Figure 7a:a point like a always allows a coalition between poor and altruistic rich agents. Figure 7bshows the one-segment, global equilibrium case.48 As Figure 7a should make clear, aRlowingthe rich to be altruistic implies that, at least at some points, they may prefer increases intargeting. Thus no intuitive restrictions are available under which we may offer a globalcharacterization of rich agents' better-than sets, nor, therefore, of the intersection of thesebetter-than sets with those of poor agents. Nonetheless, the following two lemmas allowus to characterize altruistic rich agents' better-than sets locally and to be sure that certainpoints-namely those not in poor or non-altruistic rich agents' level or better-than sets-arenot in altruistic rich agents' level or better-than sets. The lemma rules out situations likethat depicted in Figure 8.

Lemma 6 B,(k, r) C B,(kT,r) U B,(k,r).

Proof: Because V, is a convex combination of U, and Ul, it must lie between them. HIenceany point (k, ir)' E Br n B, (where Ac denotes the complement of A) must yield utility levelsU' < U, and Ul < Ul. Similarly, points in agent i's level set yield U' = Us and U. < Uj.Therefore, V, > V', so there exist £k and f, such that for a point (k',r') E L,(k,r),V,(k + Ek, r' + Cr) < Vr (k, r).Q.E.D.

The next lemma shows that our single crossing result locally withstands the introductionof altruism, so long as altruistic rich agents prefer reductions in r at the point in question.

Lemma 7 (A Local Version of the Single-Crossing Result With Altruism) Both dn/dk >dr,/dk and OV,/Or < 0 if and only if both drl/dk > d;,/dk and OVr/Ok < 0.

Proof: See Appendix 3.To understand Lemma 7, consider a point where, given no altruism on the part of the

rich, no rich-poor coalition can agree on a local increase in the degree of targeting. Thenif it is also true that the rich-given non-zero altruism-prefer a local reduction in the taxrate, two things must be true: 1) no rich-poor coalition can agree on a local increase in thedegree of targeting, even given non-zero altruism; and 2) the altruistic rich must prefer alocal reduction in the degree of targeting. Thus, at least at such points, the introduction ofaltruism does not alter the critical characteristics of rich agents' utilitiy functions, namelythat they be decreasing in both k and T and that they induce level curves that single-crosswith those of poor agents.

With the aid of an additional lemma, this result will become quite forceful. We nowshow that with when there is no targeting (k = 0), risk aversion is sufficiently small, and

4"In technical terms, we know of no general (and not terribly restrictive) conditions under which richagents' level- and better-than sets are guaranteed to be connected. Standard mathematical results guaranteethe existence, continuity, and differentiability of one part of the level set fr 'near' any given point, but theydo not rule out other such parts. We can say that non-connected level and better-than sets would requirea discrete amount of altruism, since the sets are connected with no altruism.

37

Figure 7a: The point (0,r*(O)) is alocal, but not global, equilibrium. Itcan be beaten by a coalition of poor andaltruistic rich agents, each of whomprefer the point a. This event happensonly because, as drawn, the level andbetter-than sets for altruistic agentsare not connected.

_,~ B

IP

Figure 7b: The point (O, r**(O)) is aglobal equilibrium.

(9ji

Figure 8: Altruistic rich agents' levelsets cannot look like the one drawn inthis picture. Any point in region c isworse than the point (O,r) for both poorand non-altruistic rich agents; sincealtruistic rich agents' utility is aconvex combination of these two groups'utilities, points in C must also be worsethan (0,r) for altruistic rich agents.

middle class voters are at least as altruistic as rich ones, the rich always will prefer a taxlower than that which is optimal for the middle class.

Lemma 8 (When Altruistic Rich Agents Favor Lower Taxes at (0, r*-(O))) Supposethat utilities are CRRA with p < 1, and that ,B > (>) a, i.e. rich agents weight 'own' utilityat least as heavily as do middle class ones. Then 8vIk=o = 0 implies that 9 jk=o < 0

Proof: See Appendix 3.

The condition p < 1 has reared its head again; we emphasize that this condition issufficient only, and a review of the lemma's proof should convince the reader that it isfar from necessary in general. We now can state the following proposition and theorem.Proposition 2 establishes the range of values of k and T over which no coalition betweenpoor and altruistic rich agents can defeat (0,r*`(0)); Figure 9 depicts an example of thisrange. Theorem 6 combines this result with Lemma 8 to establish that (0, r*(0)) hasequilibrium status over the full range of (k, r) values described in Proposition 2. Thefollowing definitions will be useful for the results that follow.

A(k, r) {_ (k, T) (k, r) E B,(k, T) n B,(0, r)} (35)

C*,(;) _ {(k, r): (k, r) E L,(o, f) n B,(O, ;) U Lr(O ,)} (36)

Hence A(k, r) is the set of all points preferred to (k, r) by both poor and altruistic richagents; it is a subset (not necessarily proper) of W(k, r), the win-set of (k, r) when thereis altruism.4 9 For k = 0 and tax rate ;, C,(;) is the set of all points that are exactly asgood as (0,; ) from the point of view of altruistic rich agents and no worse than (0, ;) fromthe point of view of non-altruistic rich agents.

Proposition 2 Suppose the conditions of Lemmas 8-are satisfied. Then (k,r) E C,(f)implies that neither (k, ) nor (k,r) is in A(k,r).

Proof: See Appendix 3.

Theorem 6 (Existence of Equilibrium at (0, r*'(O))) Suppose that middle class andrich agents are altruistic and that the conditions of Lemmas86hold. Then (0, T*(O)) is alocal equilibrium if C S holds.

Proof: When the results of Lemmas 4-hold, there will always be an open ball 0 such that(k,r) E C,(To-(O)) * (k,r) E O n F. lhence the set of points over which W(O, r*(O)) wasshown in the Proposition to be empty always contains some neighborhood of (0, r**(O)),which, by Lemma 5, is itself the global optimum for altruistic middle class agents.Q.E.D.

"0f course, at any point where Bm(k, r) = O, i.e. at middle class agents' optimum, it must be true thatA(k, r) = W(k, r)

38

We note that while the conditions of Lemmas 5- 8 are sufficient for the propositionand theorem, they are by no means necessary. The proposition and theorem require onlythe results of the lemmas, for which the lemmas' conditions are sufficient, not necessary.Thus there will exist parameter values violating the lemmas' conditions but still yieldingthe proposition and theorem.

For those readers who believe that altruism is an important issue in assessing transferprograms' political viability, the importance of this theorem should not be understated.It says that our model generalizes directly to the non-zero altruism case, with only minormodifications in the values other parameters must take. The single-crossing result is essen-tially unchanged, requiring only that middle class voters prefer higher taxes (when there isno targeting) than do rich voters. As we argued above, some analysts might debate such arequirement, but we think it is a natural one to make, and any argument about the issuereally seems subordinate to our primary concern, which is political support for targetingin an otherwise standard model. As for the condition under which altruistic middle classvoters will continue to favor no targeting, we do not assert that it need always be satisfiedin practice (our argument in the following subsection aside). However, we stress yet againthat the issue has been reduced to an empirical one: the objection that our model is struc-turally unable to handle altruism simply is false, leaving only arguments over the degree ofaltruism actually observed in the world.

Before turning to this question, we revisit Sri Lanka's experience, using it to calculatethe critical level of altruism above which the results of this section will not hold. RecallC 4:

a6m + (1- a)6l + (1 - a)(6i - 6)(u'(p) - 1) S °

This inequality may be restated as

a (6( - T)u'(p) + Ti-

To calculate the value of u for Sri Lanka, we take the ratio of the average income for thebottom two income quintiles to median income, which yields the value of 0.42. Assuming logutilities, we have u'(p) = I/,us. Recalling from the previous section that 61 = 0.7, Em = 0.38,and 6 = 0.43, we may calculate from the expression above that the threshold level of a is0.54, so that middle class agents would have to place a weight of 0.46 on the utility of thepoor to invalidate our conditions. Is this plausible? We think not, as we describe in thenext subsection.

7.2 Empirical Facts About Altruism and About Poverty

We begin in this subsection by arguing that the kind of altruism that is relevant for thecurrent discussion is what we call "general social, consumption-related' (GC) altruism.Since there are many types of behavior that may be construed broadly as altruistic, weemphasize that altruism need be neither consumption-related nor motivated by a desire toboost the welfare of people the altruistic actor does not know. For example, many charitableorganizations promote consumption of particular goods, irrespective of the income level or

39

preferences of the intended recipients (religious proselytizing constitutes one such example).Second, transfers may not be reciprocated immediately, even if they are part of long-tqrmimplicit insurance contracts (for example, transfers made to one's children to guaranteecare in old age). Since neither of these kinds of transfers reasonably may be interpreted asconcern for unknown beneficiaries' consumption-based standard of living, neither qualifiesas GC altruism.

We believe two facts about the world are grossly inconsistent with the hypothesis thatpeople exhibit large degrees of altruistic concern for the consumption of other citizens. First,even within families, income transfers are relatively small and do not appear to be motivatiedprimarily by altruism. Any given person i's concern for j's well-being reasonably can beexpected to decline with the social distance between the two people. Hence, estimatesof altruism-based transfers within nuclear families (i.e. between parents and children orbetween siblings) should constitute a high upper-bound on the degree of GC altruism.But intrafamilial altruism seems to be quite small. Although interhousehold transfers aregreater in some developing countries than in the U.S. (see Cox (1987] for the U.S. and Coxand Jimenez [1992] for Peru), Scheoni (1993) found that across U.S. households in 1988, theaverage transfer from parents to children was just $328. Given the small observed amountof transfers across households and the enormous differences in incomes across generations,5 0

it is no surprise that Altonji, Hayashi, and Kotlikoff (1992) easily reject the hypothesis thatthe extended family is linked altruistically. And despite large differences in consumptionacross siblings, transfers received from siblings or other relatives averaged just $44-onetenth of one percent of household income-for the U.S. in 198851. The weight of this evidencesuggests a very low degree of intrafamilial altruism.

In the face of this evidence on voluntary redistribution, we find it not credible that"general social" altruism can explain the comparatively large amount of income actuallyredistributed through the public sector. Paradoxically, we also believe that the amount ofactual income redistribution is too 8mall to admit both large degrees of GC altruism andobserved levels of poverty and deprivation; there is simply too much poverty for there toalso be much altruism.5 2 Sen's (1995) remark suggests the degree to which self-interestseems to drive observed transfers to the poor:

The political feuibility of such differential use of public services [i.e. of targeting] depends to aconsiderable extent on what the more powerful groups in a poor country see as imperative. Forexample, eaily infectious diseas receive much greater attention than other types of maladiesdo, and they tend to get eliminated with remarkable efficiency. It has happened to smallpox, nearly happened to malaria, and is on the way to happening to cholera. Even the poorwould tend to get a lot of attention partly for good humanitarian reasons, but lo becausea poor person with infectious disease is a source of infection for others. Ailments that arenot so infectious, including regular undernourishment, do not get quite that comprehensiveattention.

5 0Moreover, both Solon (1992) and Zimmerman (1992) show that the correlation of fathers' and sons'incomes in the U.S. is only about 0.4.

"t There are models suggesting that large degrees of altruism are consistent with no inter-sibling transfers,but we find such models less compelling than the simpler hypothesis of small degrees of altruim.

52 And it appears that the level of contributions by US residents to international charities supports thisview: according to Wolpert (1993), this figure was just 2.6 billion in 1991. Moreover, data in Ribar andWilhelm (1994) suggest that this number probably may weU be a high upper bound.

40

I sometimes wonder whether there is any way of making poverty terribly infectious. If that

were to happen, its general elimination would be, I am certain, remarkably rapid.

Of course, Sen's sentiment brings up an important dimension of non-altruistic concernfor the welfare of the poor. We have treated the motivation for including the welfare ofthe poor in non-poor agents' utilities as altruistically based, but there are at least threeother reasons for doing so. First, lower welfare for the poor may be asscociated withdirect welfare costs borne by the non-poor (e.g. Sen's infectious externalities). Second, thethreat of distruptive political action is likely to be related to the economic welfare of thepoor53 . Third, raising the welfare of the poor might cause beneficial spillovers to the non-poor. Some new growth models posit a positive link between threshold levels of educationand economic growth, so that higher welfare for the poor actually will raise growth rates,benefitting the non-poor as well. In light of these possible motivations, perhaps the reasonsfor direct concern by the non-poor about the welfare of the poor have nothing to do withaltruism. But proper treatment of such considerations would warrant separate models, andin any event these issues are beyond the scope of the current work.

8 Conclusion

In this paper, we have reviewed briefly some recent literature whose central theme is thatthe use of permanent income characteristic indicator targeting would allow policymakersto reduce poverty. We then developed a simple and general model, which we used to showtwo main conclusions. First, while social choice theory suggests that it is unsurprising thatpositive levels of targeted transfers almost never (in the measure-theoretic sense) can existin equilibrium, we develop simple conditions-that targeted transfers are no better for themiddle class than an actuarially fair gamble and that rich and poor agents not be ableto agree on more targeting and less taxation-under which no targeting is an equilibrium.Second, we argue that the empirical finding that introduction of targeting with a fixedbudget could greatly improve social welfare in fact has no power. this finding is perfectlyconsistent with Nash equilibrium in a game played by voters and a social-welfare maximizing(e.g. poverty minimizing) policymaker. Moreover, we give an example in which a no-targeting transfer regime satisfies the median voter's optimization problem and also locallysatisfies the policy-maker's optimization problem; a no-targeting regime can thus exist inwhat we refer to as local Nash equilibrium.

We have three further remarks. First, like all models, ours is stylized and not immune toattack. Nonetheless, we think the ball now is squarely in the other court. We have presenteda positive political economy model in which simple conditions imply that introduction ofindicator targeting will have budget-reducing effects, and in which ignoring these effectscan reduce the welfare of the poor. Even if our model is not to be accepted, proponents ofindicator targeting should be able to produce at least some plausible positive model in whichthe fixed-budget approach does lead to an optimal outcome. And of course, simply assumingthe budget is fixed counts neither as plausible nor as a model. A positive recommendation

53 Glewwe writes that "Use of dummy variables for ethnic groups helps in targeting transfers but. . . thispractice could lead, quite literally, to riots in the streets."

41

to a policymaker that she "should" target ought to be a recommendation that takes accountof constraints actually faced by that policymaker.

Second, since both types of transfers we discuss in this model are financed with aproportional tax, each is enormously redistributive. Therefore, to suggest that targeting islikely to be politically infeasible or budget-reducing is not to say that redistribution relativeto pre-transfer income distribution is infeasible. Our point is that there are practical limitsto not only the scope, but also the form of redistribution.5 4 One need not believe thatcurrent patterns of government expenditures reflect the scope limits to argue that the formconstraints bind. Just as our model's negative conclusions regarding targeting hold onlyunder certain conditions, whether a greater total reduction in poverty is feasible via moretargeting at lower budgets or no targeting at existing budgets is fundamentally an empiricalquestion. To pretend otherwise is to rely more on rhetoric than on rigor.

Third, it would be somewhat awkward to discuss a positive political economy modelwithout also discussing the positive politics of partisans on both sides of the targetingdebate. In fact, attitudes towards targeting span the political spectrum: while politicalpositions taken by partisans appear to depend on empirical beliefs about political reality,they do not seem to be related systematically to general ideological fault lines. People whocare about the poor may favor or oppose targeting, depending on their beliefs concerningthe political reaction function, r-, derived above. For those who believe that budgets areexogenous and cannot be influenced, improved targeting seems to be the right strategy toprotect the poor. To others, opposition to targeting is de ngeur: near-term implementationof targeting is expected to cause long-term erosion of political support for all redistribution.

By contrast, strongly anti-statist people (who we certainly do not mean to imply areindifferent to the plight of the poor) might easily be in favor of targeting rather thanuniveralism. Consider, for example, Deepak Lal (1994), who writes that "An implicitobjective of those who argue against targeting and in favor of univeral welfare states isdistributivist. This is not surprising as they are by and large socialists who subscribe tothe common end of egalitarianism" (our italics).

In final summary, this paper has sought to show that the policy maker in the openingdialogue might really be right. It is theoretically and empirically reasonable to believe thatattempts to achieve "more for the poor" through the use of indicator targeting may in factmean less for the poor.

54Non- lump sum-i.e. non-indicator-targeting might fare considerably better. For example, self-selectiontargeting programu make benefits universally available to anyone willing to accept certain conditions. Thusthe Nichols and Zeckhauser (1982] approach to targeting not only handles incentive compatibility issues, butmay also solve political economy problems. Public works jobs form one example of such programs: sincemiddle class agents can anticipate the availability of their benefits in times of need, they are much morelikely to support such programs. This logic might explain the apparently solid political backing enjoyed bythe Employment Guarantee Scheme in Maharashtra.

42

Appendix 1: Proofs of Results in Section 4Lem-a 2 Suppose is = 0. Then y < q, i.e. ,r < q(l - oem), if and only if

(i) r(0) < 1 and(ii) At least one of 8U,^/8r, aUm/fk < 0 at any point (k, 1)

Proof: To show (i), differentiate U.. partially with respect to r. When we evaluate at k = 0, there will be

no difference across targeting states in after-transfer income. Hence we have

1rn =pyu'(N)-q(l-y)(N + I-r) (37)

At r = 1, the u' terms drop out, so we have

py 2 q(l - y)

which is just another way of writing 7 < q. Hence utility is decreasing in r at (0,1).As for (ii), it can be shown that

aam =fmk a- m + Pit6rnU'(N + O) + (I- r)u'(N)]

- g(l.-I)[6mu'(N + O + I-r) +(1 -fn)u'(N + 1-r)] (38)

Substituting r = 1, we see that the terms multiplied by pg and q(l - g) are equal, so that this part of8U,,,/r can be non-negative only if pV > q(l - 1), which again implies y < q. Hence under the lemma'shypothesis, aUrn/ar can be non-negative only if either BUm/clk is not.

Q.E.D.Lemma 3 (Slngle Crossing for r, & n) If C 2 is satisfied, then

(i) Any two indifference curtce rt(k; Us) and r7(k; c.) intersect at most once(ii) If ri(k) = rr(k), then n(k) < (>) r,(k) V k < (>)k

For k = 0, (i) and (ii) hold only if C 2 is satisfied.Proof: We first dispense with the case when 61 < b, i.e. when C I is satisfied (note that C 2 is satisfiedtrivially in this case, since its left hand side is negative). We know that BU1/Br > 0, and from Lemma 1,we also have aUl/Bk < 0. By the Implicit Function Theorem,

dn _ BUe/lk (39)dk _Uu/8r'

so that in this case, dr,/dk > 0. We also know that the utility of the rich is decreasing in both k and

r, which implies that dr,/dk < 0 (we could derive this result algebraically by differentiation of 10 with

respect to k). Hence we have one strictly increasing function and one strictly decreasing function, and the

conclusions of the Lemma are obvious in this circumstance. We now turn to the case in which Et > 6.

First, writing out the right hand side of ( 39) after dividing by 6ly, we have

drl

dk;(k) [y(pu'(N + O) + qu'(N + O + p)} - -yspu'(N) + qu'(N + p)f]

(1 + ik)(pu'(N + O) + qu'(N + O +#)] + yn(l-k)[pu'(N) + qu'(N +pu)]

43

We will want to derive a lower bound for dn/dk > 0, so we are free to do anything to ( 40) that makesthe right hand side smaller. Thus we replace pu'(N) + qu'(N + p) with pu'(N + O) + qu'(N +O+,a) in both

the numerator and the denominator. We can do this with the numerator since this part of the numerator

is positive and pu'(N + 9) + qu'(N + 5 + A) < pu'(N) + qu'(N + p) (by concavity of u). The numerator

thus becomes

-n(k)(pu'(N + O) + qu'(N + O + p))[7 - zy],

and -y - > 0 since T < 61 by hypothesis. Since the denominator is strictly positive, the whole (modified)term must be negative. We are therefore free to reduce the value of the denominator, since reductions

in the denominator of a negative term make the term more negative, i.e. smaller. Thus we may replacepU'(N) + qu'(N + p) with pu'(N + 9) + qu'(N + O + p) in the denominator as well; but this step allows usto cancel out all terms in ( 40) involving u, leaving us with the simple result that

dn > -n(k) 7 +(1- 7 g+k (40)

This inequality holds strictly for all k > 0 and with equality at k = 0. Next, differentiating r, with

respect to k, we have

dr, yr (41)dk r -(I - k)v

Combining ( 40) and ( 41), we have dn/dk > dr./dk (with equality only at k = 0, and then only if

(42) holds with equality) if

- n(k)l +(1 -*)71+ I 7 > r r -(1 - t)y(42)

By hypothesis, we want to consider points (k,;) such that n(k) = r,(i) = r. We may thus cancel nand r, from the above expression; simplifying further, all i terms drop out, and we are left with the simplecondition that 6/61 > I - q/r, which is just condition C 2. Hence when C 2 is satisfied, n is at least as

steeply sloped as is r, at any point of intersection (and strictly more steeply sloped for all k > 0). This fact

implies that n must cross r, from below at any point of intersction with k > 0. Hence by continuity of

both n and r,, if n intersects r, at some value k of k, it must lie above r, for all k > k and vice-versa for

all k < k. Hence the second conclusion of the lemma holds; but this conclusion actually implies the first.Q.E.D.

Lemma 4 At least one of C I or C 2 is satisfied.

Proof: Rewrite C I as 6m = atbiAl/(al + o,), where the condition is satisfied iff A < 1. Then we can rewrite

C 2 as follows:

r< qe, r+ e(l -A)

or

r(e,. + at(l - A)) S q (43)

44

Violation of both conditions requires the inequality to be reversed in the above expression, as well as

A > 1. But we assumed earlier that y = ar,r + qorm < q, and if A > 1, then the left hand side of( 43) is less

than e,r < a,r + oam Hence violating both conditions induces a contradiction.

Q.E.D.

Theorem 3 Suppose that Cl > Cm. Then both C 1 and C 2 are satisfied if and only if E(s)

{(O, T(O)))

Proof: We know from Theorem 2 that no interior point of F can be an equilibrium; thus we consider only

boundary points in this proof Also, since middle class agents are risk averse, both they and the poor

support a positive level of the tax at any k. Hence we can rule out any points of the form (k, 0). We now

tackle the rest of the proof.

(Sufficiency) First, suppose that C 2 is violated. Then by Lemma 3, no point (0, r) can be a local equilibrium,

since there exist points (k, r') E W(O, r) for some k arbitrary close to zero; hence E(O) is empty for any

open ball 0 n Jr around (0, r) . Next, suppose that C I is violated. Then at k = 0, aU../ak > 0; this

fact is easily seen by differentiating Urn partially with respect to k and evaluating at k = 0. By hypothesis,

6C > Cm. > C, so that at k = 0 we also have aUllak > 0. Hence there is majority support for a local increase

in k.

We now need establish sufficiency only for points (k, 1). To rule out these points, we note first that the median

voter theorem implies that no point (k, r) with r $ r (k) can be an equilibrium. Next, by Lemma 2, if

WUm(k, l)/dr = 0, then aUl;(k, 1)/4k > 0. Hence if aU1(k, 1)/alk > 0, (k, 1) can be beaten by a poor-middle

class coalition to raise k. Then we can have equilibrium at (k, 1) only if WUt(k, 1)/8k < 0. Differentiating

U1 and U. partially with respect to k, we have

aUi(k, 1)/ak < o * 7 -y7

andalfUm(k, l)8k > o X >7m (44)

Both of these conditions hold simultaneously if and only if 61 < Cm, which violates our hypothesis.

Hence when C I is violated, the equilibrium set must be empty. Lastly, if C I is satisfied, then the rich and

the middle class both favor reductions in k whenever k > 0, so only (0,1) remains as a posible equilibrium.

But Lemma 2 established that rT(0) < 1.

Thus we have established sufficiency.

(Necessity) Suppose both conditions hold. Then by Lemma 3, no point (0, r) can be beaten by any point

with k > 0. Moreover, by the Median Voter Theorem, all points (0, r), r $ r'(0), are beaten by (0, r'(0)).

Hence (0, r'(0)) is the unique unbeaten point, establishing its solitary membership in E(r).

Q.E.D.

45

Appendix 2: Allowing positive earned income for the poorOur first step is to recaU that (IC), always wiU bind. As we noted above, positive levels of transfers

are not possible when this constraint is violated, while both the poor and the middle class will want somelevel of transfers. Hence we have two regimes to consider: r E (0, 1 -p] and r E (I - p, 1 - p/r]. While thefunctions k' and r were defined for p = 0, nothing about them is changed for r < 1 - p. That is, we mayretain the same analysis as before for the range [0,1 - p] of r' and the domain [0, 1 -p] of k'.

Next, since we build moral hazard into the model, whenever r > I - pi employed middle class workerswill take p -marginal product jobs, so that they receive untaxable earned income of p in any "high-taxregime". Hence middle clas agents' utility must be strictly increasing in r for all r E (1 - a, 1 - p/r], sothat both the middle class and the poor support a tax of 1- p/r when the high-tax regime is given. Thoughthe rich will support the lowest tax possible, just as before, the poor and middle class form a majoritycoalition, so that any high-tax regime equilibrium must occur at r = 1 - p/r.

Our method of eliminating points in the high-tax regime as possible equilibria is simple. If we canensure that the tax base always is sufficiently small in the high-tax regime, then both N and O will be lowerunder this regime than in the low-tax regime. In this case, both the poor and the middle clas-whetheremployed or not-wiU vote for a tax of 1 - p instead of I - p/r, no matter what value k takes. Thus for anyk, there will be unanimous support against any point in the high-tax regime.

The condition we seek is simply

(a,r + qom)(1-) > erv(l-

which can be rewritten as

*iem + e,,(r _ )- P

Hence if C 5 is satisfied, no point in the high-tax regime can be in E(F(p)). We note that Theorem 2holds without modification for all X C T(p) \ [0, 1] x ( 1- p), since the analysis for such subsets is unchangedby p > 0. With these facts in hand, we can state a modified version of Theorem 3 in Corollary 1.

Corollary 1 Suppose that 6a > 6m and p < p. Then C I and C 2 are satisfied if and only if (0, min[l -

p, r(0)]) E E

Proof: As before, we are entitled by Theorem 2 to ignore interior points; we now consider the boundaries ofJ.

(Sufficiency) We want to show that if (0, min(l - p, r(0)]) E E then we must satisfy both C I andC 2. First, if (0, min[l -,p, r'(0)]) E E, then we must have single-crossing, since otherwise there would existsome point (k, r) for which a rich-poor coalition would vote instead of (0, min(l -,p, r (O)]); since C 2 isnecesary for single-crossing at k = 0, it must be satisfied. Third, C I also must hold, since otherwise wehave 61, 6m > 6, and thus a poor-middle clans majority coalition exists for a local increase in k. Hence wehave established sufficiency.(Necessity) We want to show that if C I and C 2 are satisfied, then (0, min[l - p, r(0)]) E E. The argumentin the text above implies that any point in E(F)I,.o wiU beat any point in the high-tax regime, since anysuch point beats all other points in F, which includes points (k, I - u), and any of these points unanimously

46

beats all points (k, r), r > I -jp. Thus we need consider only points in the low-tax regime. If r(0) < 1 -,

the proof of Theorem 3 establishes the result. Hence we need only consider the case when r'(0) > I - p.

By the same median voter line of argument used above, (0,1 - p) beats all points (0, r), r < I - p. By

single crossing, W(O, I - p) n (0, 1] x [0, 1 - ] = 0, and 6m, < T implies the existence of a rich-middle class

coalition to defeat all (k, 1 - IA), k > 0. Hence W(O, I - i) n F(p) = 0, so (0,1 - A) E E.

Q.E.D.

The observant reader will have noticed that in the coroUary we use 'E E', whereas in conclusion (ii)

of Theorem 3, we had "(0, r'(0)) = E. This difference is not an accident. The 'lowering" of the upper

bound of the low-tax regime from unity to I - p may make a difference-though it need not. The intuitive

explanation for this fact is that, whereas Lemma 2 establishes that at a tax of unity VUm 0 0, no such fact

need hold at a lower tax. When the point (k, I -it) on the upper boundary of F(p) is a local maximum, i.e.

VUm(k, I - p) = 0, we may be able to support that point as an equilibrium in F(pu) (i.e. when the space

of alternatives is restricted to the low-tax regime) and possibly in Y as well. The following lemma lays the

groundwork for the possible existence of equilibria with positive levels of targeting on the boundary of the

low-tax regime when js > 05

Lemma 9 Suppose p = 0. Then C 1 is violated if and only if

(i) U. has a global maximum at some point (k, r), k E (0, 1], r E (0,1)

(ii) There exists at least one local maximum (k, f) E (0, 1] x (0,1) such that either VUm(k, r) = 0 or

k = I and aUn/3a > 0 at (1, ;)

Proof: (Sufficiency) If C I is satisfied, then by Lemma 1, Urn is strictly decreasing in k, violating (ii).

Moreover, atUr/ak < OVk > 0 implies that any local (and hence global) maximum occurs at k = 0,

violating (i). Hence sufficiency is established.

(Ncesssty) The first several steps in proving both conclusions are identical. First, since 6Im > 6, t9U.(O, r)/ak >

0. Second, since V < q, from Lemma 2 we know that at least one of aU ./.ak, AU . /ar = 0 at any point

(k, 1). Third, by risk aversion, aUrn(k, 0)/ar > 0. Therefore, we have ruled out all points having k = 0 and

r E 40, 1) as candidates for VUm.(k, ;) = 0 and hence for satisfying the first order conditions for a global

maximum (in each case, the conditions are violated in the direction that implies improvement is feasible).

We note that F is compact and U,, is continuous; thus Urn must take a maximum on F, proving (i). To

prove (ii), note that VUm(i, ;) = 0 must be satisfied for any interior maximum. Then we are left only with

points of the form (1, r). If aUm/alk < 0 at such points, it is feasible to improve at these points by reducing

k. Hence either VUm(k, i) = 0 or I = 1 and aUrn/ak > 0 at (1, ;)

Q.E.D.

Using this lemma, we are able to prove Proposition 3, which establishes conditions for equilibria to exist

on the upper boundary of F(p).

Proposition 3 6i > 6m > T (and thus C I is violated) if and only if any local maximum (k, ;) of UrnIao

with k > 0 is a local equilbrium for p = I - ;. Moreover, for all local equilibria (k, f), p < ' if and only

if the set over which (k, ;) has equilibrium status can be expanded to the union of the high-tax regime and

some open ball around (k, f).

Proof: (Sufficiency) We want to show that if any local maximum (k,;) of UmI1,.o with I > 0 is a local

equilbrium for A = I - ;, then 6L 2 6m > T for some i E (0, 1]. If the hypothesis holds, then either

5 5Despite the fact that we assume p = 0 in the lemma's hypothesis.

47

k = 1 or ; = 0 or we must have single-crossing (since if all of these are violated, it is possible to find a

rich-poor coalition in favor of local changes). We know that r = 0 is never optimal, and k = 1 can be

optimal only if C 1 is violated. Hence we consider points with ) < 1. If C 1 holds, then a middle class-rich

coalition favors local reductions in k. Hence C 1 must be violated; we have only to show that 6t > bim. We

know from Theorem 2 that no interior point of the low-tax regime can be a local equilibrium, so we may

look only at points on the boundary of the low-tax regime, i.e. points (k, 1 - is) with k E (0, 1).

Because these points are in the interior of F, the hypothesis that they are local maxima implies VUm 1,-o = 0.

Since we have assumed that Is = I - ;, our local maximum of Um,,,.o occurs on the boundary of the two

regimes, so that (IC)m binds. Hence the poor and the middle class have equal consumption in all states

(though the differential 6 values mean the probability of each state's occurrence varies). Under these

circumstances, an argument similar to that in the proof of Theorem 3 implies that aU/;lk < 0 if and only

if 65 < 6bm (see ( 44) ). But if aUWa/k < 0, then there exists a rich-poor coalition in favor of local reductions

in k. Hence 6, 2 6m if any local maximum is a local equilibrium, establishing sufficiency.

(Necessitv) We want to show that if 6, 2 6m > 6, there exists some i, k E (0,1) such that any local

maximum (1,;) of Um,.6,.o with k > 0 is a local equilbrium for A = I - ;. First, by Lemma 9, we know

that our hypothesis implies the existence either of an interior point with VUm = 0 or a point (1, f) where

VUm > 0. In the first case, we have a maximum on the interior of Um.IM.O, implying that VUMI,.O = 0.

Since the poor and the middle class again have equal consumption in all states. Then 6i > 6,, implies that

aU,/ak > aUm,/8k = 0. Hence no local reduction in k has a majority; no reduction in r does either, since

our position at a relative maximum of UmIJ,.O means that we are on r'(k). Nor is any local increase in

either variable possible, since the middle class are (locally) maximizing over both and the desire of the rich

to decrease either is not shared by the poor. As for an increase in r, this will move us into the high-tax

regime, which implies a discrete reduction in both N and 0. Since our definition of local equilibrium requires

only that there be some open ball for which the point is an equilibrium, we can always find a small enough

c > 0 such that no agreement between the rich and poor to increase r to r + c while reducing k will be

possible. In the second case, where k(f) = 1, the exact same argument on all fronts (except that we now

have aUm/e9k > 0) establishes the result a fortion.

As for the last comment, regarding A < j, if this does not hold, any point (k, I -pA) is beaten by (k, I -p/Ar)

via a poor-middle-class coalition. If the condition does hold, then (k, 1 - A) is unbeaten by any point in the

high-tax regime (the argument here is familiar).

Q.E.D.

Corollary 2 The asumptions of Proposition 3 hold if and only if (k, I- A) E E(Y(ip)) for some k and A,

i.e. there exists some A such that the equilibrium set is non-empty when the alternative space is restrictedto the low-tax regime. Moreover, is < if and only if (k, 1 - ip) E E, i.e. iff (k, 1 - A) is an equilibrium

over all of T.

Proof: We begin with the first claim of the corollary.

(Sufficiency) Suppose that (k, ;) E E(F(A)). By the argument in the proof of the proposition, this point

must also be a local equilbrium, and hence the appropriate conditions must hold.

(Necessity) Suppose that the assumptions of the theorem hold. Then by Lemma 9, there is some A such

that (k, 1 - A) is a global (and hence local) maximum of UmIJ,.O. By Lemma 4, violation of C 1 implies

that we must have single-crossing, so that no rich-poor coalition can defeat (k, I - A). Since this point is

a global maximum in F when p = 0, no majority coalition involving the middle class can arise to put a

low-tax regime point in W(k, I - A). Familiar arguments imply that aUt/lk > 0 > dU,/;1k, which means

48

that no coalition can stay on the low-tax regime boundary and impose a new k. Hence (k, 1 - A) must be

an equilibrium over the entire low-tax regime.

To show the second claim of the corollary, if A < ,A, then the point (k, 1-i) E W(k, r) for any (k, r) in the

high tax regime (the unanimity argument gives the result); since (k, 1 - ) was constructed above to be in

E(Y(4)), it must therefore also be in E. On the other hand, if A > p', (k, I- ) is beaten by (k, I-/r).

Hence A < p' is necessary and sufficient for the global maximum to be in E.

We note that a point (i, I - A) satisfying Corollary 2's conclusions need not be a global maximum. To

see this, take the smallest ; for which (k, ;) is a local maximum of Uml,iJ.o. Then when A = I-;, no other

maximum of U".,o is in the low-tax regime, so that Corollary 2 applies.

The results of this section notwithstanding, one shouldn't get too excited about the possibility of

supporting boundary points with k > 0. These results may be invoked only in the zero-probability event

that at such a point is just happens to be equal to 1 - r(k) at a point (k, r'(k)) that is a local maximum

of Urn.". There is therefore no reason to believe that the equilibrium set should ever contain a point with

k > 0, at least when the set of admissible values of k is all of [0,1]. That said, we stress that these positive

results concerning the viability of positive levels of targeting have as necessary conditions both the violation

of C I and on the condition that 6b > 6m : even in the extremely unlikely event that u does satisfy the

above maximizing condition, it must be the case that targeted benefits are widely enough received if they

are to be seen in any equilibrium.

56 The sense in which this event is zero-probability is quite precise: for any continuous distribution of p

having nondegenerate support over the set [0,1], the probability that any given value of p is 'drawn" is zero

by definition; since Umn will always have a countable number of maxima, the set of u values for which I - p

is a maximum has measure zero.

49

Appendix 3: Theoretical Generalization With AltruismLemma 5 (When Altruistic Middle Class Agents Oppose Targeting)

(i) Suppose C 3 is satisfied. Then al9/rn/k < 0 for k = 0.(ii) Suppose that C 4 is satisfied and that utilities are of the CRRA form, with p < 1. Then

82Vm/8kir < 0 for k = 0.Proof:

(i) At k = 0 we have

(CM) 1 I k = a[pu'(N) + qu'(N + 1 - r)]( . - 1) + (I - t)[pu'(N) + qu'(N + p)]( = - 1)

Since we want to make this expression negative, and since the terms multiplied by u'(N) and u'(N+l-r)are negative if C 3 holds, we may replace them by something smaller; 1 will be a convenient choice. 57

Reorganizing thus yields

ir [ca6,m + (1 - a)6i - 'p + 6[a(6m - 6) + (1 - a)(6/ - lu'(p + N)]q

If we then multiply through by 6, recognize that u'(p + N) < u'(p) and also add and subtract the termq(l - cr)(6i - 6), simple algebra may be used to show that C 3 is indeed a sufficient condition for negativity.

(ii) We write the crosspartial derivative as the sum A + B - C, where, using the CRRA assumption,these three terms are defined as foUows:

A py(l - p)u'(N)[a6( + (I -)bJ

B q(l - p)[a(-- - l)u'(N + 1 - r) + (1 - a)( -_)u'(p + N)]

C q[ct(= -)u- ( + 1-r) + (1-a)( l)u (p )]C 6

With p < 1, A is clearly nonpositive. To see that B is an well, again multiply through by 6, replaceU'(N + 1 - r) with unity, and add and subtract q(l - a)(6u - 6). This procedure yields C 4, establishingnonpositivity of B. C wil be negative if and only if

[&6,n + (I1- a)6s]u"(N + 1- r) + (1- a)(61- T)[pgun(N + p)u"(N + 1- r)] > Tu"(N + 1- r),

which will hold if and only if

cv6m + ( -a )61 + (1 -a)(61 ~ ts"(N +I) -1] <pu"(N + I - r

This condition differs from C 4 only by the fact that the first term in the brackets on the LHS is notu'(p). Therefore we would have our result if it could be shown that

u"(N+I-r)

Using the CRRA asumption (actually, all we really need is non-decreasing relative risk aversion andthe normalization u'(l) = 1), this condition may be written as p < -u"(N + 1 - r), which reduces evenfurther to (N + I -r)-( +) > 1, which is true since N + I - r < 1. Hence C must be strictly negative.Q.E.D.

57As risk aversion and the degree of altruism jointly approach zero, so does r"(0), so that u'(N+ I -r) willbe arbitrarily close to u'(l), which we normalize to unity. Given this fact, any condition held to be sufficientover all parameter values (with k fixed at zero) must be able to handle replacement of u'(N + 1 - r) withunity. Replacing u'(N) with unity is for mathematical convenience, and clearly adds slack to the condition,since this term is bounded below by u'(N) > u'(F) > 1. Nonetheless, the precision gained by so doing doesnot seem worth the loss in expository convenience.

50

Lemma 7 (Adaptation of the Single-Crossing Result to the Introduction of Altruism) Bothdn/dk > dr./dk and aV,/cr < 0 if and only if both dn/dk > df;/dk and aV1/ak < 0.Proof: Define a _ 6UI/ak, b _ (1 - l)aU/ak, c =_#U/ar, and d _ (1 - fl)WU1/ar. Then we can writedf./dk =-(a + b)/(c + d), dr7/dk _-a/c, and dn/dk _-bld.(Suficiency) Hence we have dn/dk > dfr/dk if and only if

b a + (

Since b, d > 0, a + b aVI/ak < 0 implies that ( 45) can hold only if c + d =-alar < 0. Simple algebramay be used to show that if c + d _aV/Or < 0, ( 45) may be reduced to -bd > -a/c, which is anotherway of writing dn/dk > drl/dk. Hence sufficiency is established.(Necessity) Start with the following two assertions: c + d < 0 and -bd > -a/c, which form the Lemma'sjoint hypothesis. Simple algebra again yields truth of ( 45); together with and b, d > O > c + d, this factnow requires that a + b _ aVr/Ok < 0, establishing necessity.Q.E.D.

Lemma S Suppose that utilities are CRRA with p < 1, and that 0 > (>)cr, i.e. rich agents weightgown' utility at least as heavily as do middle class ones. Then -v 1h.o = 0 implies that 9v; Iko < 0.Proof: Note first that aV./ar > aKV/ar reduces to

aU" - 3Ur > (a -)U(, (46)

where the RHS is non-negative. A lower bound for U1" is -u'(N + I - r)(q -y). Hence the LHS of ( 46)will be positive if

- jI)u'(r -y (r -r )) > oru'(N + 1 - r)(q - y), (47)

which can be expressed as

p(r -') >( r -r(r-Y F)Y (48)

The LHS of this inequality lies in the interval [Br/aq, oo], while the RHS lies in the interval [1, r"], sothat p < I implies that the RHS is bounded below r. Since 6 > or and r > q, the LHS is bounded above r,proving the lemma.Q.E.D.

Proposition 2 Suppose the conditions of Lemmas 6- 8 are satisfied. Then (k,r) E Cr(f) implies thatneither (k, ) nor (k, r) is in A(k, r).

Proof: Let TL(k,r) 5 L,(k, r) and let L,(k, r) be connected. Then for V,(O, r) = V, the foUowing twofunctions will exist and be continuous and differentiable over their respective domains:

r,+ : K+(7) [O, 1] (49)and

Tr: K-(V)-_ [0,l] (50)

where

K+(V) k : 3 r E [0, I]s.t. V,(k, r) = Vand '(k) > 0} (51)Or

and K- is defined analogously, but with BV,-k,`) < 0. (Note that rr+(k; V) = r, (k; V) at any point whereav,/Or = 0.)

Next, Lemmas 6 and 7 guarantees that F,, i {Ej+,-}, are wholly contained in the set B,(0,f).Suppose that (k,,r) E C,(;). Then 4(F;V) = r0 for V,(O,r) = V and i E {+,-}. Suppose now thata,(k, r.)/ar > 0, which is to say that i = + Then by Lemma 6, there exists a ri < r,(k; U,(0, f;) forwhich dV,(k, r)/lr < 0 and V,(0, ri) = V, which is to say that -r,(F; V) = r1 . By concavity of V, in itssecond argument, altruistic rich agents' utility must be decreasing in r for all r > ri when k is held at T.

51

Since V,(T,ri) = V, it therefore must be true that Vr(Tk,) < VVr > ri. But Ui(l,r) < U Uz(O,t)for all r < n (k;U), while this latter value exceeds r.(k; Ur(O, f)) (by single crossing between poor andnon-altruistic rich agents), which itself exceeds ri, proving that A(O, ;) must be empty for all points (i, r)such that there exists some r. for which (k, rO) E Cr(;).

As for points (k0 , T), the Proposition's result follows directly by defining functions F, and domains rwith i E {+,-}, noting the concavity of V. in its first argument, and proceeding as with the Tr functions.Q.E.D.

52

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